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