CHECK IF THE LINEAR IS A FACTOR OF THE GIVEN POLYNOMIAL
Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial. It is a special case of a polynomial remainder theorem.
As discussed in the introduction, a polynomial f(x) has a factor (x-a), if and only if, f(a) = 0.
Problem 1 :
Determine whether (x+1) is a factor of the following polynomials.
(i) 6x4+7x3-5x-4
(ii) 2x4+9x3+2x2+10x+15
(iii) 3x3+8x2-6x-5
(iv) x3-14x2+3x+12
(i) Solution :
6x4+7x3-5x-4
By factor theorem, if p(-1) = 0, then (x+1) is a factor of
p(x) = 6x4+7x3-5x-4
p(-1) = 6(-1)4+7(-1)3-5(-1)-4
= 6-7+5-4
p(-1) = 0
(x+1) is a factor of the given polynomial.
(ii) Solution :
2x4+9x3+2x2+10x+15
By factor theorem, if p(-1) = 0, then (x+1) is a factor of
p(x) = 2x4+9x3+2x2+10x+15
p(-1) = 2(-1)4+9(-1)3+2(-1)2+10(-1)+15
= 2-9+2-10+15
= 0
So, (x+1) is a factor of the given polynomial.
(iii) Solution :
3x3+8x2-6x-5
By factor theorem, if p(-1) = 0, then (x+1) is a factor of
p(x) = 3x3+8x2-6x-5
p(-1) = 3(-1)3+8(-1)2-6(-1)-5
= -3+8+6-5
p(-1) ≠ 0
(x+1) ix not the factor of 3x3+8x2-6x-5.
(iv) Solution :
x3-14x2+3x+12
By factor theorem, if p(-1) = 0, then (x+1) is a factor of
p(x) = x3-14x2+3x+12
p(-1) = (-1)3-14(-1)2+3(-1)+12
p(-1) = -1-14-3+12
p(-1) ≠ 0
(x+1) is not the factor of x3-14x2+3x+12.
Problem 2 :
Determine whether (x+4) is a factor of
x3 + 3x2 - 5x + 36
Solution :
By factor theorem, if p(-4) = 0, then (x+4) is a factor of
p(x) = x3 + 3x2 - 5x + 36
p(-4) = (-4)3+3(-4)2-5(-4)+36
= -64+48+20+36
p(-4) ≠ 0
(x+4) is not the factor of x3+3x2-5x+36.
Problem 3 :
Using factor theorem show that (x-1) is a factor of
4x3-6x2+9x-7
Solution :
By factor theorem, if p(1) = 0, then (x-1) is a factor of
p(x) = 4x3-6x2+9x-7
p(1) = 4(1)3-6(1)2+9(1)-7
= 4-6+9-7
p(1) = 0
(x-1) is the factor of 4x3-6x2+9x-7.
Problem 4 :
Determine whether (2x+1) is a factor of
4x3+4x2-x-1
Solution :
By factor theorem, if p(-1/2) = 0, then (2x+1) is a
Factor of p(x) = 4x3+4x2-x-1
p(-1/2) = 4(-1/2)3+4(-1/2)2-(-1/2)-1
= -1/2+1+1/2-1
p(-1/2) = 0
(2x+1) is the factor of 4x3+4x2-x-1.
Problem 5 :
Determine the value of p if (x+3) is a factor of
x3-3x2-px+24
Solution :
By factor theorem, if p(-3) = 0, then (x+3) is a
factor of p(x) = x3-3x2-px+24
p(-3) = (-3)3-3(-3)2- p(-3)+24
This implies that -27-27+3p+24 = 0
-30 + 3p = 0
3p = 30
p = 10
So, the value of p is 10.
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