RD Sharma solutions for Mathematics for Class 9 chapter 6 - Factorisation of Polynomials [Latest edition]

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

3x² - 4x +15

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

`y^2 +2sqrt3`

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

`3sqrtx+sqrt2x`

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

`x - 4/x`

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

`x^12+y^3+t^50`

Write the coefficient of x² in the following:

`17 -2x + 7x^2`

Write the coefficient of x² in the following:

`9-12x +X^3`

Write the coefficient of x² in the following:

`pi/6x^2- 3x+4`

Write the coefficient of x² in the following:

`sqrt3x-7`

Write the degrees of each of the following polynomials

`7x3 + 4x2 – 3x + 12`

Write the degrees of the following polynomials:

`12-x+2x^3` 

Write the degrees of the following polynomials:

`5y-sqrt2`

Write the degrees of the following polynomials:

7

Write the degrees of the following polynomials

0

 Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`x+x^2 +4`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`3x-2`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`2x+x^2`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`3y`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`t^2+1`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials

`7t^4+4t^3+3t-2`

Classify the following polynomials as polynomials in one-variable, two variables etc:

`x^2-xy+7y^2`

Classify the following polynomials as polynomials in one-variable, two variables etc:

`x^2-2tx+7t^2-x+t`

Classify the following polynomials as polynomials in one-variable, two variables etc:

`t^3_3t^2+4t-5`

Classify the following polynomials as polynomials in one-variable, two variables etc:

`xy+yx+zx`

Identify polynomials in the following:

`f(x)=4x^3-x^2-3x+7`

Identify polynomials in the following:

`g(x)=2x^3-3x^2+sqrtx-1`

Identify polynomials in the following:

`p(x)=2/3x^3-7/4x+9`

Identify polynomials in the following:

`q(x)=2x^2-3x+4/x+2`

Identify polynomials in the following:

`h(x)=x^4-x^(3/2)+x-1`

Identify polynomials in the following:

`f(x)=2+3/x+4x`

 Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

`f(x)=0`

 Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

`g(x)=2x^3-7x+4`

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

`h(x)=-3x+1/2`

Identify constant, linear, quadratic and cubic polynomials from the following polynomials

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

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

`q(x)=4x+3`

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

`r(x)=3x^2+4x^2+5x-7`

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

If `f(x)=2x^2-13x^2+17x+12` find `f(2)`

If `f(x)=2x^2-13x^2+17x+12` find `f-(3)`

If `f(x)=2x^2-13x^2+17x+12` find `f(0)`

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`f ( x ) = 3x +1, x = - 1/3`

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`f(x)=x^2- 1,x=1,-1`

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`g(x)=3x^2-2,`  `x=2/sqrt3     2/sqrt3`

Verify whether the indicated numbers is zeros of the polynomials corresponding to them in the following case:

\[p(x) = x^3 - 6 x^2 + 11x - 6, x = 1, 2, 3\]

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`f ( x ) = 5x - pi , x = 4/5`

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`f ( x) = x^2and x = 0`

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`f(x) = lx + m , x = - m/1`

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

`f (x) = 2x +1, x = 1/2`

If `x = 2` is a root of the polynomial `f(x) = 2x<sup>2</sup> – 3x + 7a` find the value of a.

If `x = −1/2` is a zero of the polynomial `p(x)=8x^3-ax^2 -+2` find the value of a.

If x = 0 and x = −1 are the roots of the polynomial f(x) =2x³ − 3x² + ax + b, find the value of a and b.

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the result by actual division: (1−8)

f(x) = x³ + 4x² − 3x + 10, g(x) = x + 4

f(x) = 2x⁴ − 6x³ + 2x² − x + 2, g(x) = x + 2

f(x) = 9x³ − 3x² + x − 5, g(x) = \[x - \frac{2}{3}\]

\[f(x) = 3 x^4 + 2 x^3 - \frac{x^2}{3} - \frac{x}{9} + \frac{2}{27}, g(x) = x + \frac{2}{3}\]

Find the remainder when x³ + 3x² + 3x + 1 is divided by  x + 1.

Find the remainder when x³ + 3x² + 3x + 1 is divided by \[x - \frac{1}{2}\].

Find the remainder when x³ + 3x² + 3x + 1 is divided by x.

Find the remainder when x³ + 3x² + 3x + 1 is divided by \[x + \pi\] .

Find the remainder when x³ + 3x² + 3x + 1 is divided by 5 + 2x .

The polynomials ax³ + 3x² − 3 and 2x³ − 5x + a when divided by (x − 4) leave the remainders R₁ and R₂ respectively. Find the value of the following case, if R₁ = R_2.

The polynomials ax³ + 3x² − 3 and 2x³ − 5x + a when divided by (x − 4) leave the remainders R₁ and R₂ respectively. Find the values of the following case, if  R₁ + R₂ = 0.

The polynomials ax³ + 3x² − 3 and 2x³ − 5x + a when divided by (x − 4) leave the remainders R₁ and R₂ respectively. Find the values of the following cases, if 2R₁ − R₂ = 0.

f(x) = 3x⁴ + 17x³ + 9x² − 7x − 10; g(x) = x + 5

If x − 2 is a factor of the following two polynomials, find the values of a in each case x³ − 2ax² + ax − 1.

If x − 2 is a factor of the following two polynomials, find the values of a in each case x⁵ − 3x⁴ − ax³ + 3ax² + 2ax + 4.

In the following two polynomials, find the value of a, if x − a is  factor x⁶ − ax⁵ + x⁴ − ax³ + 3x − a + 2.

In the following two polynomials, find the value of a, if x − a is  factor  (x⁵ − a²x³ + 2x + a + 1).

In the following two polynomials, find the value of a, if x + a is a factor x³ + ax² − 2x +a + 4.

In the following two polynomials, find the value of a, if x + a is a factor x⁴ − a²x² + 3x −a.

Using factor theorem, factorize each of the following polynomials:
x³ + 6x² + 11x + 6

x³ + 2x² − x − 2

x³ − 6x² + 3x + 10

x⁴ − 7x³ + 9x² + 7x − 10

x³ − 23x² + 142x − 120

2y³ + y² − 2y − 1

Factorize of the following polynomials:

x³ + 13x² + 31x − 45 given that x + 9 is a factor

Factorize of the following polynomials:

4x³ + 20x² + 33x + 18 given that 2x + 3 is a factor.

If \[x = \frac{1}{2}\] is a zero of the polynomial f(x) = 8x³ + ax² − 4x + 2, find the value of a.

Mark the correct alternative in each of the following:
If x − 2 is a factor of x² + 3ax − 2a, then a =

If x³ + 6x² + 4x + k is exactly divisible by x + 2, then k =

If x − a is a factor of x³ −3x²a + 2a²x + b, then the value of b is

If x¹⁴⁰ + 2x¹⁵¹ + k is divisible by x + 1, then the value of k is

If x + 2 is a factor of x² + mx + 14, then m =

If x − 3 is a factor of x² − ax − 15, then a =

If x⁵¹ + 51 is divided by x + 1, the remainder is

If x + 1 is a factor of the polynomial 2x² + kx, then k =

If x + a is a factor of x⁴ − a²x² + 3x − 6a, then a =

The value of k for which x − 1 is a factor of 4x³ + 3x² − 4x + k, is

If x + 2  and x − 1 are the factors of x³ + 10x² + mx + n, then the values of m and n are respectively

Let f(x)  be a polynomial such that \[f\left( - \frac{1}{2} \right)\] = 0, then a factor of f(x) is

When x³ − 2x² + ax − b is divided by x² − 2x − 3, the remainder is x − 6. The values of a and b are respectively

One factor of x⁴ + x² − 20 is x² + 5. The other factor is

If (x − 1) is a factor of polynomial f(x) but not of g(x) , then it must be a factor of

(x+1) is a factor of xⁿ + 1 only if

If x² + x + 1 is a factor of the polynomial 3x³ + 8x² + 8x + 3 + 5k, then the value of k is

If (3x − 1)⁷  = a₇x⁷ + a₆x⁶ + a₅x⁵ +...+ a₁x + a₀, then a₇ + a₅ + ...+a₁ + a₀ =

If both x − 2 and \[x - \frac{1}{2}\] are factors of px² + 5x + r, then

If x² − 1 is a factor of ax⁴ + bx³ + cx² + dx + e, then