Question

Factorise
${x}^{3 } + {13x}^{2} + 32x + 20$
​

Answer 1

x³+13x²+32x+20

=(x+1)(x²+12x+20) [∵, for x=-1, x³+13x²+32x+20=-1+13-32+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)

HAPPY RAKSHABANDHAN TO ALL MY SISTERS

Answer 2

Let p(x) = x3 + 13x2 + 32x + 20 ± 2, ± 4, ± 5 …… By trial method

, p(−1) = (−1)3 + 13(−1)2 + 32(−1) + 20 = − 1 +13 − 32 + 20 = 33 − 33 = 0

As p(−1) is zero, therefore, x + 1 is a factor of this polynomial p(x).

Let us find the quotient on dividing x3 + 13x2 + 32x + 20 by (x + 1).

By long division,

refer the attachment for long division

It is known 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)