Question
Using factor theorem, factorize the following polynomials:
x3 + 13x2 + 32x + 20
x3 + 13x2 + 32x + 20
✓
Answer
Let p(x) = x3 + 13x2 + 32x + 20
The factors of 20 are $\pm1,\pm2,\pm4,\pm5\dots$
By hit and trial method
p(-1) = (-1)3 + 13(-1)2 + 32(-1) + 20
= -1 + 13 - 32 + 20
= 33 - 33 = 0
As p(-1) is zero, so x + 1 is a factor of this polynomial p(x).
Let us find the quotient while dividing x3 + 13x2 + 32x + 20 by (x + 1)
By long division
We know that
Dividend = Divisor × Quotient + Remainder
x3 + 13x2 + 32x + 20 = (x + 1)(x2 + 12x + 20) + 0
= (x + 1)(x2 + 10x + 2x + 20)
= (x + 1)[x(x + 10) + 2(x + 10)]
= (x + 1)(x + 10)(x + 2)
= (x + 1)(x + 2)(x + 10)
The factors of 20 are $\pm1,\pm2,\pm4,\pm5\dots$
By hit and trial method
p(-1) = (-1)3 + 13(-1)2 + 32(-1) + 20
= -1 + 13 - 32 + 20
= 33 - 33 = 0
As p(-1) is zero, so x + 1 is a factor of this polynomial p(x).
Let us find the quotient while dividing x3 + 13x2 + 32x + 20 by (x + 1)
By long division
We know that
Dividend = Divisor × Quotient + Remainder
x3 + 13x2 + 32x + 20 = (x + 1)(x2 + 12x + 20) + 0
= (x + 1)(x2 + 10x + 2x + 20)
= (x + 1)[x(x + 10) + 2(x + 10)]
= (x + 1)(x + 10)(x + 2)
= (x + 1)(x + 2)(x + 10)
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