RD Sharma 2020 solution class 9 chapter 6 Factorization of polynomial Expressions Exercise 6.5

Exercise 6.5

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Question 1:

Using factor theorem, factorize each of the following polynomials:

x³ + 6x² + 11x + 6
 

Answer 1:

Let f(x) = x³ + 6x² + 11x + 6 be the given polynomial.
Now, put the we get

Therefore, is a factor of f(x).
Now,
$f\left(x\right)=x^3+5x^2+x^2+5x+6x+6$

Hence, are the factors of f(x).

Question 2:

x³ + 2x² − x − 2

Answer 2:

Let be the given polynomial.
Now, put the x = -1, we get
$$\begin{gathered} f(-1)={\left(-1\right)}^3+2{\left(-1\right)}^2-\left(-1\right)-2 \\ =0 \end{gathered}$$
Therefore, is a factor of polynomial f(x).
Now, x³ + 2x² − x − 2 can be written as,
$f\left(x\right)=x^3+3x^2-x^2-3x+2x-2$

Hence, are the factors of the polynomial f(x).

Question 3:

x³ − 6x² + 3x + 10

Answer 3:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,
$f\left(x\right)=x^3-7x^2+x^2+10x-7x+10$

Hence, are the factors of the polynomial f(x).

Question 4:

x⁴ − 7x³ + 9x² + 7x − 10

Answer 4:

Let be the given polynomial.
Now, putting we get
$$\begin{gathered} f\left(1\right)={\left(1\right)}^4-7{\left(1\right)}^3+9{\left(1\right)}^2+7\left(1\right)-10 \\ =1-7+9+7-10=0 \end{gathered}$$
Therefore, is a factor of polynomial f(x).
Now,
$f\left(x\right)=x^4-x^3-6x^3+6x^2+3x^2-3x+10x-10$

Where
Putting we get

Therefore, is a factor of g(x).
Now,
$g\left(x\right)=x^3-7x^2+x^2-7x+10x+10$

From equation (i) and (ii), we get

Hence are the factors of polynomial f(x).

Question 5:

3x³ − x² − 3x + 1

Answer 5:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

Question 6:

x³ − 23x² + 142x − 120

Answer 6:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

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Question 7:

y³ − 7y + 6

Answer 7:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(y).
Now,

Hence are the factors of polynomial f (y).

Question 8:

x³ − 10x² − 53x − 42

Answer 8:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

Question 9:

y³ − 2y² − 29y − 42

Answer 9:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(y).
Now,

Hence are the factors of polynomial f(y).

Question 10:

2y³ − 5y² − 19y + 42

Answer 10:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(y).
Now,

Hence are the factors of polynomial f(y).

Question 11:

x³ + 13x² + 32x + 20

Answer 11:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

Question 12:

x³ − 3x² − 9x − 5

Answer 12:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

Question 13:

2y³ + y² − 2y − 1

Answer 13:

Let be the given polynomial.
Now, putting we get

Now,

Hence are the factors of polynomial f(y).

Question 14:

x³ − 2x² − x + 2

Answer 14:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

Question 15:

Factorize each of the following polynomials:

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

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

Answer 15:

(i) Let be the given polynomial.
is a factor of the polynomial f(x).
Now,

Hence are the factors of polynomial f(x).
(ii) Let be the given polynomial.
Therefore is a factor of the polynomial f(x).
Now,

Hence are the factors of polynomial f(x).

Question 16:

x⁴ − 2x³ − 7x² + 8x + 12

Answer 16:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,
$f\left(x\right)=x^4-3x^3+x^3-3x^2-4x^2+12x-4x+12$

Where
Putting we get

Therefore, is the factor of g(x).
Now,
$g\left(x\right)=x^3-2x^2-x^2-6x+2x+12$

From equation (i) and (ii), we get

Hence are the factors of polynomial f(x).

Question 17:

x⁴ + 10x³ + 35x² + 50x + 24

Answer 17:

Let $f\left(x\right)=x^4+10x^3+35x^2+50x+24$ be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Where
Putting we get

Therefore, is the factor of g(x).
Now,

From equation (i) and (ii), we get

Hence are the factors of polynomial f(x).

Question 18:

2x⁴ − 7x³ − 13x² + 63x − 45

Answer 18:

Let be the given polynomial.
Now, putting we get

Therefore, is a factor of polynomial f(x).
Now,

Where
Putting we get

Therefore, is a factor of g(x).
Now,

From equation (i) and (ii), we get

Hence are the factors of polynomial f(x).