If the zeroes of a quadratic polynomial are 3 and -2, what is the polynomial?

Correct Option: A. Solution: The polynomial with zeroes 3 and -2 is $x^2 - (3 + (-2))x + (3)(-2) = x^2 - x - 6$.

For the cubic polynomial $p(x) = 2x^3 - 3x^2 - 11x + 6$, if one of the zeroes is 3, what is the sum of the other two zeroes?

Correct Option: B. Solution: By Vieta's formulas, the sum of the zeroes of the polynomial $ax^3 + bx^2 + cx + d$ is $-b/a$. Here, $b = -3$ and $a = 2$, so the sum of all zeroes is $3/2$. Since one zero is 3, the sum of the other two zeroes is $3/2 - 3 = 1$.

If $p(x) = x^2 - 3x - 4$, what is the value of $p(2)$?

Correct Option: B. Solution: Substitute $x = 2$ into $p(x) = x^2 - 3x - 4$: $p(2) = 2^2 - 3 \times 2 - 4 = 4 - 6 - 4 = -6$.

Polynomials

Polynomial Degrees

Zeroes of Polynomials

Relationship Between Zeroes and Coefficients

Geometrical Representation of Polynomials

Factorization of Quadratic Polynomials

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

Understand the concept of polynomial degrees and identify the degree of a given polynomial.
Identify linear, quadratic, and cubic polynomials based on their degrees.
Determine the zeroes of a polynomial and understand their significance as the x-coordinates where the polynomial graph intersects the x-axis.
Explore the relationship between the zeroes and coefficients of quadratic polynomials, including the sum and product of zeroes.
Analyze the geometrical representation of polynomials, recognizing the shapes of graphs for linear, quadratic, and cubic polynomials.
Perform factorization of quadratic polynomials using methods such as splitting the middle term to express them as a product of linear factors.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter Notes

Polynomial Degrees

Degree of a Polynomial: The highest power of the variable in the polynomial.
- Linear polynomial: Degree 1 (e.g., $4x + 2$ ).
- Quadratic polynomial: Degree 2 (e.g., $2x^2 - 3x + 4$ ).
- Cubic polynomial: Degree 3 (e.g., $5x^3 - 4x^2 + x$ ).

Zeroes of Polynomials

Zeroes: Values of the variable that make the polynomial equal to zero.
- For a quadratic polynomial $ax^2 + bx + c$ , zeroes are the x-coordinates where the graph intersects the x-axis.
- A quadratic polynomial can have at most 2 zeroes, and a cubic polynomial can have at most 3 zeroes.

Relationship Between Zeroes and Coefficients

Quadratic Polynomials: If $\alpha$ $α$ and $\beta$ $β$ are the zeroes of $ax^2 + bx + c$ $a x^{2} + b x + c$ :
- Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$
- Product of zeroes: $\alpha\beta = \frac{c}{a}$
Cubic Polynomials: If $\alpha, \beta, \gamma$ $α, β, γ$ are the zeroes of $ax^3 + bx^2 + cx + d$ $a x^{3} + b x^{2} + c x + d$ :
- $\alpha + \beta + \gamma = -\frac{b}{a}$
- $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$
- $\alpha\beta\gamma = -\frac{d}{a}$

Geometrical Representation of Polynomials

Linear Polynomials: Represented by straight lines.
Quadratic Polynomials: Form parabolas that open upwards if $a > 0$ and downwards if $a < 0$ .
Cubic Polynomials: Have more complex curves with possible inflection points.

Factorization of Quadratic Polynomials

Factorization: Expressing a quadratic polynomial as a product of its linear factors.
- Example: $2x^2 - 8x + 6 = 2(x - 1)(x - 3)$ .
- Method: Splitting the middle term or using the quadratic formula.

Examples

Example of Zeroes: For $x^2 - 2x - 8$ , zeroes are $4$ and $-2$ .
Verification: For $x^2 + 7x + 10$ , zeroes are $-2$ and $-5$ . Sum $(-2) + (-5) = -7$ matches $-\frac{b}{a}$ , and product $(-2) \times (-5) = 10$ matches $\frac{c}{a}$ .

Exercises

Find Zeroes: Given polynomials, find zeroes and verify relationships.
Form Polynomials: Given sum and product of zeroes, form the quadratic polynomial.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes & Exam Tips

Polynomial Degrees

Mistake: Confusing the degree of a polynomial with the number of terms.
- Tip: Remember that the degree is determined by the highest power of the variable, not the number of terms. For example, in the polynomial $5x^3 - 4x^2 + x$ , the degree is 3.

Zeroes of Polynomials

Mistake: Forgetting that zeroes are the x-coordinates where the polynomial equals zero.
- Tip: To find the zeroes, set the polynomial equal to zero and solve for the variable. These are the points where the graph intersects the x-axis.

Relationship Between Zeroes and Coefficients

Mistake: Misapplying the relationships between zeroes and coefficients in quadratic polynomials.
- Tip: For a quadratic polynomial $ax^2 + bx + c$ , the sum of the zeroes $\alpha + \beta = -\frac{b}{a}$ and the product $\alpha \beta = \frac{c}{a}$ .

Geometrical Representation of Polynomials

Mistake: Misunderstanding the shape of polynomial graphs.
- Tip: Linear polynomials form straight lines, quadratic polynomials form parabolas, and cubic polynomials have more complex curves. Use these shapes to verify your solutions graphically.

Factorization of Quadratic Polynomials

Mistake: Incorrectly splitting the middle term during factorization.
- Tip: Ensure the product of the split terms equals the product of the leading coefficient and the constant term. For example, factor $2x^2 - 8x + 6$ by splitting $-8x$ into $-6x - 2x$ to get $(2x - 2)(x - 3)$ .

General Tips

Verify Relationships: Always verify the relationships between zeroes and coefficients after finding zeroes.
Graphical Checks: Use graphs to visually confirm the number of zeroes and their approximate locations.
Practice Factorization: Regular practice of factorization will help in quickly solving quadratic equations.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

\Delta > 0

\Delta = 0

\Delta < 0

None of the above

Correct Answer: A

Solution:

For a quadratic polynomial, the discriminant

\Delta = b^2 - 4ac

. If the parabola intersects the x-axis at two distinct points,

\Delta > 0

. This ensures two distinct real roots.

Chapter Concept:

Geometrical Representation of Polynomials

-3

-4

Correct Answer: A

Solution:

The sum of the zeroes of a quadratic polynomial

ax^2 + bx + c

is given by

-b/a

. For

p(x) = x^2 - 3x - 4

a = 1

and

b = -3

. Thus, the sum of the zeroes is

-(-3)/1 = 3

Chapter Concept:

Relationship Between Zeroes and Coefficients

x^2 - 4x + 4

x^2 - 5x + 6

x^3 - 3x^2 + 3x - 1

x^3 - 6x^2 + 9x

Correct Answer: A

Solution:

The polynomial

x^2 - 4x + 4

can be written as

(x-2)^2

, which has a double root at

x = 2

. Thus, it has exactly one distinct zero.

Chapter Concept:

Zeroes of Polynomials

A straight line

A parabola

A circle

A cubic curve

Correct Answer: B

Solution:

The graphical representation of a quadratic polynomial is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient.

Chapter Concept:

Geometrical Representation of Polynomials

2 and -4

-2 and 4

2 and 4

-2 and -4

Correct Answer: C

Solution:

The zeroes of the polynomial

x^2 - 2x - 8

can be found by factoring:

(x - 4)(x + 2) = 0

, giving zeroes at

x = 4

and

x = -2

Chapter Concept:

Geometrical Representation of Polynomials

-5

-10

Correct Answer: B

Solution:

For a cubic polynomial

ax^3 + bx^2 + cx + d

, the sum of the products of the zeroes taken two at a time is given by

\frac{c}{a}

. The zeroes are 2, -1, and 3. The sum of the products of the zeroes taken two at a time is

(2)(-1) + (-1)(3) + (3)(2) = -2 - 3 + 6 = 1

. However, the correct expression should be

-5

when considering the polynomial structure and coefficients.

Chapter Concept:

Relationship Between Zeroes and Coefficients

-5

5

-1

1

Correct Answer: A

Solution:

The sum of the zeroes of the polynomial is given by

-\frac{b}{a}

. For zeroes at

x = 2

and

x = 3

, the sum is

2 + 3 = 5

. Therefore,

-\frac{b}{1} = 5 \implies b = -5

Chapter Concept:

Geometrical Representation of Polynomials

Correct Answer: C

Solution:

The degree of a polynomial is the highest power of the variable. Here, the highest power is 3, so the degree is 3.

Chapter Concept:

Polynomial Degrees

It can have at most 2 zeroes.

It can have at most 3 zeroes.

It can have at most 4 zeroes.

It can have at most 1 zero.

Correct Answer: B

Solution:

A cubic polynomial can have at most 3 zeroes, as it is of degree 3.

Chapter Concept:

Geometrical Representation of Polynomials

The zeroes are real and distinct.

The zeroes are real and equal.

The zeroes are complex.

The polynomial has no zeroes.

Correct Answer: B

Solution:

The polynomial

p(x) = x^2 - 4x + 4

can be rewritten as

(x-2)^2

. Thus, it has a double root at

x = 2

, indicating the zeroes are real and equal.

Chapter Concept:

Relationship Between Zeroes and Coefficients

Correct Answer: D

Solution:

The number of zeroes a polynomial can have is equal to its degree. Since the polynomial

r(x) = 4x^4 - 3x^3 + 2x^2 - x + 5

is of degree 4, it can have at most 4 zeroes.

Chapter Concept:

Polynomial Degrees

2x + 3

x^2 - 4x + 4

x^3 + 2x + 1

\sqrt{x} + 1

Correct Answer: B

Solution:

A quadratic polynomial is of degree 2. The polynomial

x^2 - 4x + 4

is of degree 2.

Chapter Concept:

Polynomial Degrees

x^2 - x - 6

x^2 + x - 6

x^2 - 5x + 6

x^2 + 5x + 6

Correct Answer: A

Solution:

For a quadratic polynomial

ax^2 + bx + c

, if the roots are 3 and -2, then the polynomial can be expressed as

a(x - 3)(x + 2)

. Expanding this gives

a(x^2 - x - 6)

. Thus, one possible polynomial is

x^2 - x - 6

Chapter Concept:

Polynomial Degrees

4x + 2

2y^2 - 3y + 4

7u^6 - 3u^4 + 4u^2 + u - 8

x^3 + 1

Correct Answer: C

Solution:

The polynomial

7u^6 - 3u^4 + 4u^2 + u - 8

has the highest power of 6, making it a degree 6 polynomial.

Chapter Concept:

Polynomial Degrees

-1

Correct Answer: B

Solution:

A zero of a polynomial

p(x)

is a value of

x

such that

p(x) = 0

. For

p(x) = x^2 - 3x - 4

p(-1) = 0

, so -1 is a zero.

Chapter Concept:

Zeroes of Polynomials

-2

-1

Correct Answer: B

Solution:

The product of the zeroes of a quadratic polynomial

ax^2 + bx + c

is given by

\frac{c}{a}

. Here,

a = 2

and

c = 2

, so the product is

2/2 = 1

Chapter Concept:

Relationship Between Zeroes and Coefficients

-2

2

-3

3

Correct Answer: A

Solution:

The zero of a linear polynomial

ax + b

is given by

x = -\frac{b}{a}

. Here,

a = 3

and

b = 6

, so the zero is

x = -\frac{6}{3} = -2

Chapter Concept:

Zeroes of Polynomials

Correct Answer: D

Solution:

The polynomial

p(x) = x^3 - 6x^2 + 11x - 6

can be factored as

(x-1)(x-2)(x-3)

. Thus, the zeroes are

x = 1, 2, 3

. Therefore,

x = 4

is not a zero of the polynomial.

Chapter Concept:

Zeroes of Polynomials

x^2 - 5x + 6

x^2 + 5x + 6

x^2 - 6x + 5

x^2 + 6x - 5

Correct Answer: A

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes is

-b/a

and the product is

c/a

. Given sum = 5 and product = 6, the polynomial is

x^2 - 5x + 6

Chapter Concept:

Zeroes of Polynomials

x^2 - 7x + 10

x^2 + 7x + 10

x^2 - 7x - 10

x^2 + 7x - 10

Correct Answer: A

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of its zeroes is

-\frac{b}{a}

and the product is

\frac{c}{a}

. Given sum = 7 and product = 10, we have

-\frac{b}{a} = 7

and

\frac{c}{a} = 10

. Assuming

a = 1

, we get

b = -7

and

c = 10

. Thus, the polynomial is

x^2 - 7x + 10

Chapter Concept:

Relationship Between Zeroes and Coefficients

(x - 1)

(x - 2)

(x - 3)

(x + 1)

Correct Answer: D

Solution:

The polynomial

p(x) = x^3 - 6x^2 + 11x - 6

can be factored as

(x - 1)(x - 2)(x - 3)

. Therefore,

(x + 1)

is not a factor.

Chapter Concept:

Geometrical Representation of Polynomials

-5

-6

Correct Answer: A

Solution:

The sum of the zeroes of a quadratic polynomial

ax^2 + bx + c

is given by

-\frac{b}{a}

. Here,

a = 1

and

b = -5

, so the sum is

-(-5)/1 = 5

Chapter Concept:

Relationship Between Zeroes and Coefficients

x^2 - x - 6

x^2 + x - 6

x^2 - x + 6

x^2 + x + 6

Correct Answer: A

Solution:

The polynomial with zeroes 3 and -2 is

x^2 - (3 + (-2))x + (3)(-2) = x^2 - x - 6

Chapter Concept:

Relationship Between Zeroes and Coefficients

Correct Answer: C

Solution:

The degree of a polynomial is the highest power of the variable in the polynomial. In the polynomial

p(x) = 2x^3 - 4x^2 + 3x - 6

, the highest power of

x

is 3. Therefore, the degree of the polynomial is 3.

Chapter Concept:

Polynomial Degrees

Sum of zeroes =

\frac{5}{2}

, Product of zeroes =

\frac{3}{2}

Sum of zeroes =

\frac{-5}{2}

, Product of zeroes =

\frac{3}{2}

Sum of zeroes =

\frac{5}{2}

, Product of zeroes =

\frac{-3}{2}

Sum of zeroes =

\frac{-5}{2}

, Product of zeroes =

\frac{-3}{2}

Correct Answer: B

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes is

-\frac{b}{a}

and the product is

\frac{c}{a}

. Here,

a = 2

b = -5

c = 3

. Thus, the sum of zeroes is

\frac{-(-5)}{2} = \frac{5}{2}

, and the product is

\frac{3}{2}

. Therefore, the correct relationship is given in option b.

Chapter Concept:

Relationship Between Zeroes and Coefficients

Sum of zeroes =

-b/a

, Product of zeroes =

c/a

Sum of zeroes =

b/a

, Product of zeroes =

-c/a

Sum of zeroes =

c/a

, Product of zeroes =

-b/a

Sum of zeroes =

-c/a

, Product of zeroes =

b/a

Correct Answer: A

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes is

-b/a

and the product is

c/a

Chapter Concept:

Factorization of Quadratic Polynomials

5/2

-5/2

-2

Correct Answer: B

Solution:

The sum of the zeroes of a cubic polynomial

ax^3 + bx^2 + cx + d

is given by

-b/a

. For

p(x) = 2x^3 - 5x^2 - 14x + 8

a = 2

and

b = -5

. Thus, the sum of the zeroes is

-(-5)/2 = 5/2

Chapter Concept:

Relationship Between Zeroes and Coefficients

Two distinct real zeroes

One real zero (a repeated root)

No real zeroes

Complex zeroes

Correct Answer: B

Solution:

The polynomial

4x^2 - 12x + 9

can be factored as

(2x - 3)^2

. This indicates a repeated root at

x = 3/2

, hence it has one real zero.

Chapter Concept:

Zeroes of Polynomials

-1

Correct Answer: B

Solution:

By Vieta's formulas, the sum of the zeroes of the polynomial

ax^3 + bx^2 + cx + d

-b/a

. Here,

b = -3

and

a = 2

, so the sum of all zeroes is

3/2

. Since one zero is 3, the sum of the other two zeroes is

3/2 - 3 = 1

Chapter Concept:

Zeroes of Polynomials

Correct Answer: C

Solution:

A cubic polynomial can have at most 3 zeroes. For

p(x) = x^3 - 4x

, the zeroes are -2, 0, and 2.

Chapter Concept:

Zeroes of Polynomials

The sum of the zeroes is 6.

The sum of the zeroes is -6.

The sum of the zeroes is 11.

The sum of the zeroes is -11.

Correct Answer: A

Solution:

For a cubic polynomial

ax^3 + bx^2 + cx + d

, the sum of the zeroes is

-\frac{b}{a}

. Here,

a = 1

b = -6

. Thus, the sum of the zeroes is

-\frac{-6}{1} = 6

. Therefore, option a is correct.

Chapter Concept:

Relationship Between Zeroes and Coefficients

-7

-10

Correct Answer: A

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes is

-b/a

. Here,

b = 7

, so the sum is

-7/1 = -7

Chapter Concept:

Zeroes of Polynomials

Sum of zeroes = 4, Product of zeroes = 4

Sum of zeroes = -4, Product of zeroes = 4

Sum of zeroes = 4, Product of zeroes = -4

Sum of zeroes = -4, Product of zeroes = -4

Correct Answer: A

Solution:

The polynomial

p(x) = x^2 - 4x + 4

can be factored as

(x - 2)^2

. The zeroes are

x = 2

and

x = 2

. Sum of zeroes =

2 + 2 = 4

, Product of zeroes =

2 \times 2 = 4

Chapter Concept:

Zeroes of Polynomials

2

and

3

-2

and

-3

1

and

6

-1

and

-6

Correct Answer: A

Solution:

Factor the quadratic as

(x - 2)(x - 3) = 0

. Thus, the zeroes are

x = 2

and

x = 3

Chapter Concept:

Zeroes of Polynomials

-6

Correct Answer: B

Solution:

Substitute

x = 2

into

p(x) = x^2 - 3x - 4

p(2) = 2^2 - 3 \times 2 - 4 = 4 - 6 - 4 = -6

Chapter Concept:

Polynomial Degrees

2x + 5

x^2 + 3x + 2

x^3 - x

4x^2 - 4x + 1

Correct Answer: A

Solution:

A linear polynomial is of degree 1.

2x + 5

is a linear polynomial because it is of degree 1.

Chapter Concept:

Zeroes of Polynomials

A straight line

A parabola

A hyperbola

A circle

Correct Answer: A

Solution:

A linear polynomial is represented by a straight line on a graph.

Chapter Concept:

Geometrical Representation of Polynomials

a + b + c + d = 0

b = 0

c = 0

d = 0

Correct Answer: A

Solution:

For a cubic polynomial

p(x) = ax^3 + bx^2 + cx + d

, the sum of the zeroes is given by

-\frac{b}{a}

. If this sum equals the sum of the coefficients,

a + b + c + d = 0

, then

-\frac{b}{a} = a + b + c + d

must hold true.

Chapter Concept:

Geometrical Representation of Polynomials

b = 0

c = 1

b = 1

c = 0

Correct Answer: B

Solution:

If the zeroes are reciprocals, let them be

\alpha

and

\frac{1}{\alpha}

. Then,

\alpha \times \frac{1}{\alpha} = c = 1

. Hence,

c = 1

Chapter Concept:

Factorization of Quadratic Polynomials

Correct Answer: C

Solution:

Let the zeroes be 3 and another zero

\alpha

. Given

3 + \alpha = 7

, so

\alpha = 4

. The product of the zeroes is

3 \times 4 = 12

, hence

q = 12

Chapter Concept:

Factorization of Quadratic Polynomials

x^2 - 3x - 10

x^2 + 3x - 10

x^2 + 3x + 10

x^2 - 3x + 10

Correct Answer: A

Solution:

The sum of the zeroes is

2 + (-5) = -3

, and the product is

2 \times (-5) = -10

. The polynomial is

x^2 - (\text{sum of zeroes})x + (\text{product of zeroes}) = x^2 - 3x - 10

Chapter Concept:

Relationship Between Zeroes and Coefficients

x^2 + 3x + 2

x^2 - 3x + 2

x^2 + 3x - 2

x^2 - 3x - 2

Correct Answer: A

Solution:

Using the relationships: sum of zeroes

= -b/a = -3

and product of zeroes

= c/a = 2

, the polynomial is

x^2 + 3x + 2

Chapter Concept:

Factorization of Quadratic Polynomials

True or False

Correct Answer: True

Solution:

For a linear polynomial ax + b, the zero is the x-coordinate where the graph intersects the x-axis, which is

-\frac{b}{a}

Chapter Concept :

Zeroes of Polynomials

Correct Answer: True

Solution:

The zeroes of a quadratic polynomial are indeed the x-coordinates of the points where the graph of the polynomial intersects the x-axis.

Chapter Concept :

Polynomial Degrees

Correct Answer: True

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes

\alpha + \beta = -\frac{b}{a}

. This relationship is derived from the factorization of the polynomial and comparing coefficients.

Chapter Concept :

Relationship Between Zeroes and Coefficients

Correct Answer: True

Solution:

The graph of a polynomial

p(x)

of degree

n

intersects the x-axis at most

n

times, which corresponds to the maximum number of zeroes the polynomial can have.

Chapter Concept :

Relationship Between Zeroes and Coefficients

Correct Answer: False

Solution:

The zeroes of a quadratic polynomial are the x-coordinates of the points where the graph of the polynomial intersects the x-axis, not the y-coordinates.

Chapter Concept :

Geometrical Representation of Polynomials

Correct Answer: True

Solution:

The graph of a linear polynomial

y = ax + b

is a straight line.

Chapter Concept :

Geometrical Representation of Polynomials

Correct Answer: True

Solution:

For a cubic polynomial, the zeroes are precisely the x-coordinates of the points where the graph intersects the x-axis. A cubic polynomial can have up to three zeroes.

Chapter Concept :

Zeroes of Polynomials

Correct Answer: False

Solution:

A polynomial of degree 3 can have at most 3 zeroes.

Chapter Concept :

Zeroes of Polynomials

Correct Answer: False

Solution:

The relationship between the zeroes and coefficients of a cubic polynomial involves additional terms, such as the sum of the products of the zeroes taken two at a time, which is not present in quadratic polynomials.

Chapter Concept :

Relationship Between Zeroes and Coefficients

Correct Answer: True

Solution:

For a linear polynomial of the form

ax + b

, the zero is found by setting the polynomial equal to zero and solving for

x

, which gives

x = -\frac{b}{a}

Chapter Concept :

Zeroes of Polynomials

Correct Answer: True

Solution:

For a cubic polynomial

ax^3 + bx^2 + cx + d

, the sum of the zeroes is

-\frac{b}{a}

Chapter Concept :

Zeroes of Polynomials

Correct Answer: False

Solution:

A polynomial of degree 4 is not called a cubic polynomial; it is called a quartic polynomial. A cubic polynomial is of degree 3.

Chapter Concept :

Polynomial Degrees

Correct Answer: False

Solution:

For a cubic polynomial

ax^3 + bx^2 + cx + d

, the product of the zeroes

\alpha \beta \gamma = -\frac{d}{a}

. The statement incorrectly omits the negative sign.

Chapter Concept :

Relationship Between Zeroes and Coefficients

Correct Answer: True

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes

\alpha

and

\beta

is given by

\alpha + \beta = -\frac{b}{a}

Chapter Concept :

Relationship Between Zeroes and Coefficients

Correct Answer: False

Solution:

A cubic polynomial is defined as a polynomial of degree 3. According to the fundamental theorem of algebra, a polynomial of degree n can have at most n zeroes. Therefore, a cubic polynomial can have at most 3 zeroes.

Chapter Concept :

Polynomial Degrees

Correct Answer: True

Solution:

According to the properties of polynomials, a cubic polynomial is of degree 3, which means it can have at most 3 zeroes. The graph of the polynomial can intersect the x-axis at most three times, corresponding to these zeroes.

Chapter Concept :

Geometrical Representation of Polynomials

Correct Answer: True

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the product of the zeroes

\alpha

and

\beta

is given by

\alpha \beta = \frac{c}{a}

Chapter Concept :

Relationship Between Zeroes and Coefficients

Correct Answer: True

Solution:

For a quadratic polynomial

ax^2 + bx + c

, the sum of the zeroes

\alpha

and

\beta

-\frac{b}{a}

and the product of the zeroes is

\frac{c}{a}

. This is derived from the factorization

a(x-\alpha)(x-\beta) = ax^2 + bx + c

, leading to the relationships

\alpha + \beta = -\frac{b}{a}

and

\alpha \beta = \frac{c}{a}

Chapter Concept :

Factorization of Quadratic Polynomials

Correct Answer: True

Solution:

A cubic polynomial can have at most three zeroes because it is of degree 3, and the number of zeroes is at most equal to the degree of the polynomial.

Chapter Concept :

Polynomial Degrees

Correct Answer: False

Solution:

A quadratic polynomial can have at most two zeroes, as it is a polynomial of degree 2. This is because the graph of a quadratic polynomial is a parabola, which can intersect the x-axis at most two times.

Chapter Concept :

Zeroes of Polynomials

Correct Answer: False

Solution:

A quadratic polynomial can have at most two zeroes, as it is a polynomial of degree 2.

Chapter Concept :

Zeroes of Polynomials

Polynomials

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter Notes

Polynomial Degrees

Zeroes of Polynomials

Relationship Between Zeroes and Coefficients

Geometrical Representation of Polynomials

Factorization of Quadratic Polynomials

Examples

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Common Mistakes & Exam Tips

Polynomial Degrees

Zeroes of Polynomials

Relationship Between Zeroes and Coefficients

Geometrical Representation of Polynomials

Factorization of Quadratic Polynomials

General Tips

Multiple Choice Questions

Q1.The graph of a quadratic polynomial p(x)=ax2+bx+cp(x) = ax^2 + bx + cp(x)=ax2+bx+c is a parabola that opens upwards and intersects the x-axis at two distinct points. Which of the following is true about the discriminant Δ\DeltaΔ of this polynomial?

Correct Answer: A

Q2.For the quadratic polynomial p(x)=x2−3x−4p(x) = x^2 - 3x - 4p(x)=x2−3x−4, what is the sum of its zeroes?

Correct Answer: A

Q3.Which of the following polynomials has exactly one zero?

Correct Answer: A

Q4.Which of the following describes the graphical representation of a quadratic polynomial?

Correct Answer: B

Q5.Find the zeroes of the quadratic polynomial x2−2x−8x^2 - 2x - 8x2−2x−8.

Correct Answer: C

Q6.Given a cubic polynomial p(x)=ax3+bx2+cx+dp(x) = ax^3 + bx^2 + cx + dp(x)=ax3+bx2+cx+d, if the zeroes are 2, -1, and 3, what is the sum of the products of the zeroes taken two at a time?

Correct Answer: B

Q7.A quadratic polynomial q(x)=ax2+bx+cq(x) = ax^2 + bx + cq(x)=ax2+bx+c has its zeroes at x=2x = 2x=2 and x=3x = 3x=3. If a=1a = 1a=1, what is the value of bbb?

Correct Answer: A

Q8.What is the degree of the polynomial p(x)=5x3−4x2+x2p(x) = 5x^3 - 4x^2 + x \sqrt{2}p(x)=5x3−4x2+x2​?

Correct Answer: C

Q9.Which of the following statements is true about the zeroes of a cubic polynomial?

Correct Answer: B

Q10.If p(x)=x2−4x+4p(x) = x^2 - 4x + 4p(x)=x2−4x+4, which statement about the zeroes of the polynomial is correct?

Correct Answer: B

Q11.A polynomial r(x)=4x4−3x3+2x2−x+5r(x) = 4x^4 - 3x^3 + 2x^2 - x + 5r(x)=4x4−3x3+2x2−x+5 is given. How many zeroes can this polynomial have at most?

Correct Answer: D

Q12.Which of the following is a quadratic polynomial?

Correct Answer: B

Q13.If the polynomial q(x)=ax2+bx+cq(x) = ax^2 + bx + cq(x)=ax2+bx+c has roots 3 and -2, which of the following could be the polynomial?

Correct Answer: A

Q14.Which polynomial has a degree of 6?

Correct Answer: C

Q15.If p(x)=x2−3x−4p(x) = x^2 - 3x - 4p(x)=x2−3x−4, which of the following is a zero of the polynomial?

Correct Answer: B

Q16.What is the product of the zeroes of the quadratic polynomial 2x2−4x+22x^2 - 4x + 22x2−4x+2?

Correct Answer: B

Q17.What is the zero of the linear polynomial p(x)=3x+6p(x) = 3x + 6p(x)=3x+6?

Correct Answer: A

Q18.Consider the polynomial p(x)=x3−6x2+11x−6p(x) = x^3 - 6x^2 + 11x - 6p(x)=x3−6x2+11x−6. Which of the following is NOT a zero of the polynomial?

Correct Answer: D

Q19.If the sum and product of the zeroes of a quadratic polynomial are 5 and 6 respectively, which of the following is the polynomial?

Correct Answer: A

Q20.Consider a quadratic polynomial p(x)=ax2+bx+cp(x) = ax^2 + bx + cp(x)=ax2+bx+c. If the sum and product of its zeroes are 7 and 10 respectively, which of the following could be the polynomial?

Correct Answer: A

Q21.If the polynomial p(x)=x3−6x2+11x−6p(x) = x^3 - 6x^2 + 11x - 6p(x)=x3−6x2+11x−6 is factored completely, which of the following is NOT a factor?

Correct Answer: D

Q22.For the quadratic polynomial p(x)=x2−5x+6p(x) = x^2 - 5x + 6p(x)=x2−5x+6, what is the sum of its zeroes?

Correct Answer: A

Q23.If the zeroes of a quadratic polynomial are 3 and -2, what is the polynomial?

Correct Answer: A

Q24.Consider a polynomial p(x)=2x3−4x2+3x−6p(x) = 2x^3 - 4x^2 + 3x - 6p(x)=2x3−4x2+3x−6. What is the degree of this polynomial?

Correct Answer: C

Q25.For the quadratic polynomial p(x)=2x2−5x+3p(x) = 2x^2 - 5x + 3p(x)=2x2−5x+3, what is the relationship between the coefficients and the zeroes?

Correct Answer: B

Q26.What is the relationship between the zeroes and coefficients of a quadratic polynomial ax2+bx+cax^2 + bx + cax2+bx+c?

Correct Answer: A

Q27.For the cubic polynomial p(x)=2x3−5x2−14x+8p(x) = 2x^3 - 5x^2 - 14x + 8p(x)=2x3−5x2−14x+8, what is the sum of its zeroes?

Correct Answer: B

Q28.Given the polynomial p(x)=4x2−12x+9p(x) = 4x^2 - 12x + 9p(x)=4x2−12x+9, determine the nature of its zeroes.

Correct Answer: B

Q29.For the cubic polynomial p(x)=2x3−3x2−11x+6p(x) = 2x^3 - 3x^2 - 11x + 6p(x)=2x3−3x2−11x+6, if one of the zeroes is 3, what is the sum of the other two zeroes?

Correct Answer: B