Question

what must be subtracted from x^3-6x^2-15x+80 so that the result is exactly divisible by x^2+x-12??

Answer 1

Let p (x) = x3 - 6x2 - 15x + 80 and q (x) = x2 + x - 12

By division algorithm, when p (x) is divided by q (x), the remainder is a linear expression in x.

So, let r (x) = ax + b is subtracted to p (x) so that p (x) + r (x) is divisible by q (x).

Let, f (x) = p (x) – r (x)

⇒ f(x) = x3 - 6x2 - 15x + 80 – (ax + b)

⇒ f(x) = x3 - 6x2 – (a + 15)x + (80 – b)

We have,

q(x) = x2 + x – 12

⇒ q(x) = (x + 4) (x - 3)

Clearly, q (x) is divisible by (x + 4) and (x - 3) i.e. (x + 4) and (x - 3) are factors of q (x)

Therefore, f (x) will be divisible by q (x), if (x + 4) and (x - 3) are factors of f (x).

i.e. f(-4) = 0 and f(3) = 0

f (3) = 0

⇒ (3)3 – 6(3)2 – 3 (a + 15) + 80 – b = 0

⇒ 27 – 54 – 3a – 45 + 80 – b = 0

⇒ 8 – 3a – b = 0 (i)

f (-4) = 0

⇒ (-4)3 – 6 (-4)2 – (-4) (a + 15) + 80 – b = 0

⇒ -64 – 96 + 4a + 60 + 80 – b = 0

⇒ 4a – b – 20 = 0 (ii)

Subtract (i) from (ii), we get

⇒ 4a – b – 20 – (8 – 3a – b) = 0

⇒ 4a – b – 20 – 8 + 3a + b = 0

⇒ 7a = 28

⇒ a = 4

Put value of a in (ii), we get

⇒ b = -4

Putting the value of a and b in r (x) = ax + b, we get

r (x) = 4x – 4

Hence, p (x) is divisible by q (x), if r (x) = 4x – 4 is subtracted from it.

Answer 2

x³ – 6x² – 15x + 80 will be divisible by x² + x − 12, if (4x - 4) is - subtracted from it.

Given,

x³-6x²-15x+80 is the dividend and x²+x-12 is the divisor

To Find,

Find the number that must be subtracted from x³-6x²-15x+80 so that the result is exactly divisible by x²+x-12

Solution,

By divisible algorithm, when

p(x) = x³ – 6x² – 15x + 80 is divided by x² + x - 12 the reminder is a linear polynomial

Let r(x) = a(x) + b be subtracted from p(x) so that the result is divisible by q(x).

Let, f(x) = p(x) - q(x)

= x³ 6x²-15x+80 − (ax + b)

= x³ = 6x² - (a+15)x + 80 - b

We have,

q(x) = x² + x - 12

= x² + 4x - 3x - 12

= (x+4)(x − 3)

Clearly, (x+4) and (x-3)are factors of q(x), therefore, f(x) will be divisible by q(x) if (x+4) and (x-3) are factors of f(x), i.e. f (-4) and f (3) are equal to zero.

Therefore,

ƒ(−4) = (−4)³ – 6(−4)² – (a + 15)(-4) +80-b = 0

or, 64-96+4a+60+80-b=0

or, -20+4a-b= 0

or, 4a-b= 20____________(1)

and, f(3) = (3)³- (3)² - (a+15)(3) +80-b=0

or, 27-54-3a-45+ 80-b=0

or, -8-3a-b =0

or, 3a+b = 8 __________(2)

From, 1 and 2 we get, a=4 and b=-4

Hence, x³ – 6x² – 15x + 80 will be divisible by x² + x − 12, if (4x - 4) is - subtracted from it.

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