Question

7 To factorise x3 + 13x2 + 32x + 20, we
(a) split the middle term
(b) combine x3 + 13x2 and 32x + 20 (C) combine x3 + 32x and 13x2 + 20 (d) use factor theorem to find factors.

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Answer 1

Required Answer:-

Solution:

To factorise x³ + 13x² + 32x + 20, we need to,

1. Split the middle term. ❌
2. Combine x³ + 13x² and 32x + 20 ❌
3. Combine x³ + 32x and 13x² + 20 ❌
4. Use factor theorem to find factors. ✔

Let's factorise it.

Let,

f(x) = x³ + 13x² + 32x + 20 ......(i)

★ Putting x = -1, we get,

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

= -1 + 13 - 32 + 20

= 33 - 33

= 0

Therefore,

➡ f(-1) = 0

So, by factor theorem, x + 1 is a factor.

★ On dividing (x³ + 13x² + 32x + 20) by (x + 1), we get (x² + 12x + 20) as quotient and 0 as remainder.

Therefore, the other factors of f(x) are factors of x² + 12x + 20

So,

x³ + 13x² + 32x + 20

= (x + 1)[x² + 12x + 20]

= (x + 1)[x² + 10x + 2x + 20]

= (x + 1)[x(x + 10) + 2(x + 20)]

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

Hence, factorised form of the polynomial is (x + 1)(x + 2)(x + 10)

★ Hence, Option 4 is the right answer.

Another method to factorise this,

x³ + 13x² + 32x + 20

= x³ + x² + 12x² + 12x + 20x + 20

= x²(x + 1) + 12x(x + 1) + 20(x + 1)

= (x + 1)(x² + 12x + 20)

= (x + 1)[x² + 2x + 10x + 20]

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

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

Hence, factorised form is (x + 1)(x + 2)(x + 10).