Question

Factorize
Please solve it​

Answer 1

Correct question :

Factorise the foIIowing :

x³ + 13x² + 32x + 20

Given :

• x³ + 13x² + 32x + 20

SoIution :

We can factorise it in two methods i.e. first one through factorisation method . Second one through factor theorem.

Let's do it through factor theorem.

Let f(x) = x³ + 13x² + 32x + 20

∴ f(-1) = (-1)³ + 13(-1)² + 32(-1) + 20

         = -1 + 13 × 1 - 32 + 20

         = -1 + 13 - 12

         = 12 - 12

         = 0

According to factor thorem, x - (-1) ⇔ (x + 1) is a factor of f(x)

Now factorising that expression :

⇒ x³ + 13x² + 32x + 20

⇒ x³ + x² + 12x² + 12x + 20x + 20

⇒ x²(x + 1) + 12x(x  + 1) + 20(x + 1)

⇒ (x² + 12x + 20)(x + 1)

⇒ (x² + 10x + 2x + 20)(x + 1)

⇒ {x(x + 10) + 2(x + 10)}(x + 1)

⇒ (x + 10) (x + 2) (x + 1)

⇒ (x + 1)(x + 2)(x + 10)   ············· (Answer)

Answer 2

Answer:

Let p(x) = x3 + 13x2 + 32x + 20

p(-1) = -1 + 13 - 32 + 20 = -33 + 33 = 0

Therefore (x + 1) is a factor of p(x).

On dividing p(x) by (x + 1) we get

p(x)  (x + 1) = x2 + 12x + 20

Thus,

x3 + 13x2 + 32x + 20 = (x + 1)(x2 + 12x + 20)

= (x + 1) (x2 + 10x + 2x + 20)

= (x + 1)[x(x + 10) + 2(x + 10)]

= (x + 1) (x +2) (x + 10)

Hence, x3 + 13x2 + 32x + 20 = (x + 1) (x +2) (x + 10).

2nd method

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)

                             

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