# Factor Theorem Examples And Solutions

Factor theorem (गुणनखंड प्रमेय)

If p(x) is a polynomial of degree n ≥ 1 and a is a factor of p(x) then (i) (x – a) is a factor of p(x), if p(a)=0 and (ii) p(a)=0, if x – a is a factor of p(x).

Factor Theorem Examples And Solutions

Example 1:-

Determine which of the following polynomials has (x + 2) a factor?

बताइए कि निम्नलिखित बहुपदों में से किस बहुपद का एक गुणनखंड (x + 2) है?

Explanation:

By factor theorem we know that if ( x – a) is a factor if p(x) then p(a) must be equal to zero. So for the following polynomials if p(-2) is equal to zero then it will be a factor of p(x)

A) p(x) = x²+5x+6

p(-2) = (-2)²+5(-2)+6 = 4 – 10 + 6 = 10 – 10 = 0

Here p(-2) = 0 so x + 2 is a factor of this polynomial.

B)  p(x) = x²+3x-10

p(-2) = (-2)²+3(-2)-10 = 4 – 6 – 10 = 4 – 16 = -12

For this polynomial p(-2) is not equal to zero so x + 2 is not a factor of this polynomial.

C) p(x) =x²+7x+10

p(-2) = (-2)²+7(-2)+10 = 4-14+10 = 14-14 = 0

Here p(-2) = 0 so x + 2 is a factor of this polynomial.

Example 2:-

Using Factor theorem determine whether (x + 5) is a factor of  x²+7x+10?

गुणनखंड प्रमेय लागू करके बताइए कि (x + 5), x²+7x+10 का एक गुणनखंड है या नहीं?

Explanation:

p(x) = x²+7x+10

By factor theorem we know that if (x + 5) is a factor if p(x) then p(-5) must be zero.

p(-5) = (-5)²+7(-5)+10 = 25 – 35 + 10 = 35 – 35 = 0

So x + 5 is a factor of p(x).

Example 3:-

Determine whether y + 4 is a factor of  (3y)²+18y+24?

गुणनखंड प्रमेय लागू करके बताइए कि (y + 4), (3y)²+18y+24  का एक गुणनखंड है या नहीं?

Explanation:

p(y) = (3y)²+18y+24

By factor theorem we know that if (y + 4) is a factor if p(y) then p(-4) must be zero.

p(-4) = 3(-4)²+18(-4)+24

= (3×16) + (18×4)+ 24

= 48 – 72 + 24 = 72 – 72 = 0

So y + 4 is a factor of p(y).

Example 4:-

Find the value of k if x – 2 is a factor of  x²-x+k?

K का मान पता कीजिए यदि (x – 2), x²-x+k का एक गुणनखंड है?

Explanation:

p(x) = x²-x+k

If (x – 2) is a factor of p(x) then p(2) must be equal to zero.

p(2) = 2²-2+k

0 = 4 – 2 + k

0 = 2 + k

k = -2

Example 5:-

Find the value of k if x + 1 is a factor of  x²-3x+k?

k का मान पता कीजिए यदि (x + 1), x²-3x+k  का एक गुणनखंड है?

Explanation:

p(x) = x²-3x+k

If x + 1 is a factor of p(x) then p(-1) must be equal to zero.

p(-1) = (-1)²-3(-1)+k

0 = 1 + 3 + k

0 = 4 + k

k = -4

Example 6:-

Find the value of k if x – 2 is a factor of  kx²+x-10

k का मान पता कीजिए यदि (x – 2), kx²+x-10 का एक गुणनखंड है?

Explanation:

p(x) = kx²+x-10

If x – 2 is a factor of p(x) then p(2) must be zero.

p(2) = k(2)²+2-10

0 = 4k +2 – 10

0 = 4k – 8

k = 2