RD Sharma 2020 solution class 9 chapter 6 Factorization of polynomial Expressions Exercise 6.2

Exercise 6.2

page-6.8

Question 1:

If f(x) = 2x³ − 13x² + 17x + 12, find (i) f(2) (ii) f(−3) (iii) f(0)

Answer 1:

Let be the given polynomial

(i) The value of f (2) can be found by putting x = 2

        = 10 

(ii) The value of f (–3) can be found by putting x = –3

(iii) The value of f (0) can be found by putting x = 0

Question 2:

Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:

(i) $f\left(x\right)=3x+1, x=-\frac{1}{3}$

(ii) $f\left(x\right)=x^2-1, x=1, -1$

(iii) $g\left(x\right)=3x^2-2, x=\frac{2}{\sqrt{3}}, -\frac{2}{\sqrt{3}}$

(iv) $p\left(x\right)=x^3-6x^2+11x-6, x=1, 2,3$

(v) $f\left(x\right)=5x-\mathrm{\pi }, \mathrm{x}=\frac{4}{5}$

(vi) $f\left(x\right)=x^2, x=0$

(vii) $f\left(x\right)=lx+m, x=-\frac{m}{1}$

(viii) $f\left(x\right)=2x+1, x=\frac{1}{2}$

Answer 2:

(i) To check whether the given number is the zero of the polynomial or not we have to find

Hence, is the zeros of the given polynomial.

(ii) To check whether the given number is the zero of the polynomial or not we have to find and

Now,

And ,

Hence, and x = 1 are the zeros of the polynomial.

(iii) To check whether the given number is the zero of the polynomial or not we have to find

Now,

And

Here, both the value of , are not satisfied the polynomial. Therefore, they are not the zeros of the polynomial.

(iv) To check whether the given number is the zero of the polynomial or not we have to find

And

Hence, x = 1, 2, 3 are the zeros of the polynomial p(x).

(v)To check whether the given number is the zero of the polynomial or not we have to find

Here, , does not satisfy therefore, is not a zero of the polynomial.

(vi)To check whether the given number is the zero of the polynomial or not we have to find

Hence, x = 0 is the zeros of the polynomial.

(vii) To check whether the given number is the zero of the polynomial or not we have to find

Hence, is zeros of the polynomial.

(viii)To check whether the given number is the zero of the polynomial or not we have to find

Hence, does not satisyf the polynomial, so, is not zero of the polynomial.

Question 3:

If x = 2 is a root of the polynomial f(x) = 2x² − 3x + 7a, find the value of a.

Answer 3:

The given polynomial is

If x = 2 is the root of the polynomial.

Then

Question 4:

If $x=-\frac{1}{2}$ is a zero of the polynomial p(x) = 8x³ − ax²−x + 2, find the value of a.

Answer 4:

The given polynomial is

If is a zeros of the polynomial p(x).

then

Therefore,

Hence the value of

Question 5:

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.

Answer 5:

The given polynomial is

f(x) =2x³ − 3x² + ax + b

If is zeros of the polynomial f(x), then f(0) = 0

Similarly, if x = − 1 is the zeros of the polynomial of,

Then, 

Putting the value of b from equation (1)

Thus,

Question 6:

Find the integral roots of the polynomial f(x) = x³ + 6x² + 11x + 6.

Answer 6:

The given polynomial is

Here, f(x) is a polynomial with integer coefficient and the coefficient of highest degree term is 1. So, the integer roots of f(x) are factors of 6. Which are by observing.

          = 0   

Also,

And similarly,

f(−3) = 0

Therefore, the integer roots of the polynomial f(x) are −1, −2, − 3

Question 7:

Find rational roots of the polynomial f(x) = 2x³ + x² − 7x − 6.

Answer 7:

The given polynomial is

f(x) is a cubic polynomial with integer coefficients. If $\frac{b}{c}$ is rational root in lowest terms, then the values of b are limited
to the factors of 6 which are $\pm 1,\pm 2,\pm 3,\pm 6$ and the values of c are limited to the factor of 2 as $\pm 1,\pm 2$. Hence, the possible
rational roots are $\pm 1,\pm 2,\pm 3,\pm 6,\pm \frac{1}{2},\pm \frac{3}{2}$. 

Since,

So, 2 is a root of the polynomial

Now, the polynomial can be written as,

Also,

Therefore,

Hence, the rational roots of the polynomialare 2, – 3/2 and – 1.