# Class 10 R S AGGARWAL AND V AGGARWAL Solutions Maths Chapter 2 - Polynomials

## Polynomials Exercise Ex. 2A

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We have

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Let

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### Solution 4

2x^{2} - x
- 6 = 0

⇒ 2x^{2} -
4x + 3x - 6 = 0

⇒ 2x(x - 2) + 3(x - 2) = 0

⇒ (x - 2)(2x + 3) = 0

⇒ (x - 2) = 0 or (2x + 3) = 0

So, the zeroes of the given quadratic polynomial are 2 and

Sum of the zeros

Product of the zeros

### Solution 11

5x^{2} +
10x = 0

⇒ 5x(x + 2) = 0

⇒ 5x = 0 or (x + 2) = 0

⇒ x = 0 or x = -2

So, the zeroes of the given quadratic polynomial are 0 and -2.

Sum of the zeros

Product of the zeros

### Solution 13

p(x) = 2x^{2}
+ 5x + k

Here, a = 2, b = 5 and c = k

α + β = and αβ =

α^{2} + β^{2}
+ αβ = …. Given

### Solution 17

Let α = 2 and β = -6

Sum of zeros = α + β = 2 + (-6) = -4

Product of zeros = αβ = 2(-6) = -12

So, the required quadratic polynomial is

x^{2} - (α + β)x + αβ = x^{2} + 4x - 12

Here, a = 1, b = 4 and c = -12

Sum of the zeros

Product of the zeros

### Solution 19

## Polynomials Exercise Ex. 2B

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### Solution 10

Here, remainder = 2x + 3 = px + q

⇒ p = 2 and q = 3

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1

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### Solution 21

f(x) = 3x^{3}
+ 16x^{3} + 15x - 18

is one zero of f(x).

is a factor of f(x).

⇒ (3x - 2) is also a factor of f(x).

### Solution 22

Here, f(x) = 2x^{4}
- 3x^{3} - 3x^{2} + 6x - 2

1 and are two zeros of f(x).

⇒ is a factor of f(x).

⇒is a factor of f(x).

### Solution 18

### Solution 19

Here, f(x) = x^{4}
+ x^{3} - 14x^{2} - 2x + 24

Since, √2 and -√2 are two zeros of f(x)

⇒ (x - √2) and (x + √2) is a factor of f(x).

⇒ (x - √2) and (x + √2) = x - 2 is a factor of f(x).

### Solution 20

Here, f(x) = 2x^{4}
- 13x^{3} + 19x^{2} + 7x - 3

Let α = (2 + √3) and β = (2 - √3). Then,

α + β = (2 + √3).+ (2 - √3) = 4

αβ = (2 + √3).(2 - √3) =4 - 3 = 1

So, the quadratic polynomial whose roots are α and β is given by

x^{2} - (α
+ β)x + αβ
= x^{2} - 4x + 1

⇒ x^{2}
- 4x + 1 is a factor of f(x).

### Solution 23

## Polynomials Exercise Ex. 2C

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### Solution 17

If f(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can always find polynomials q(x) and r(x) such that f(x) = q(x)g(x) + r(x),

where r(x) = 0 or degree r(x) < degree g(x).

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## Polynomials Exercise MCQ

### Solution 1

Correct answer: (d)

An expression of the form p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ….. + a_{n}x^{n}, where a_{n} ≠ 0, is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, ……, a_{n} are real numbers and each power of x is a non-negative integer.

### Solution 2

Correct answer: (d)

An expression of the form p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ….. + a_{n}x^{n}, where a_{n} ≠ 0, is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, ……, a_{n} are real numbers and each power of x is a non-negative integer.

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## Polynomials Exercise FA

### Solution 1

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