Question

If x+2 and x-1 are factors of x^3+10x^2+mx+n,find m and n?
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Answer 1

Answer:

m = 7 and n = –18

Step-by-step explanation:

Given polynomial,

x³ + 10x² + mx + n

(x + 2) is a factor of the polynomial.

x + 2 = 0

x = –2

Put x = –2, the result is zero.

(–2)³ + 10(–2)² + m(–2) + n = 0

–8 + 10(4) – 2m + n = 0

–8 + 40 – 2m + n = 0

2m – n = 32

n = 2m – 32

(x – 1) is a factor of the polynomial.

x – 1 = 0

x = +1

Put x = +1, the result is zero.

(1)³ + 10(1)² + m(1) + n = 0

1 + 10 + m + n = 0

m + n = –11

Put n = 2m – 32,

m + 2m – 32 = –11

3m = –11 + 32

3m = 21

m = 21/3 = 7

Value of n :

n = 2m – 32

n = 2(7) – 32

n = 14 – 32

n = –18

Answer 2

Question :-

If x+2 and x-1 are factors of x^3+10x^2+mx+n,find m and n?

Answer :-

• m = 7
• n = -18

Step by step explanation :-

Given :-

• If x+2 and x-1 are factors of x³ + 10x² + mx + n

We know that,

x + 2 and x - 1 are the factors also can be written as,

• x = -2
• x = 1

Substituting x = - 2 in the given equation,

$\rm:\longmapsto{( - 2) {}^{3} + 10( - 2) {}^{2} + m( - 2) + n = 0} \\ \\ \rm:\longmapsto{ - 8 + 10(4) - 2m + n = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto{ - 8 + 40 = 2m - n} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto \red{2m - n = 32 \: \: \: \: \: \: \: \: \: \: \: - - - (1) \: \: \: \: \: \: \: }$

Substituting x = 1 in the given equation,

$\rm:\longmapsto{(1) {}^{3} + 10(1) {}^{2} + m(1) + n = 0 } \\ \\\rm:\longmapsto{1 + 10 + m + n = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto \red{m + n = - 11} \: \: \: \: \: \: \: \: - - - (2)$

Adding equation (1) and equation (2)

$\rm:\longmapsto{2m - n + (m + n) = 32 + ( - 11)} \\ \\ \rm:\longmapsto{2m + m - n + n = 21} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto{3m = 21} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto{m = \frac{21}{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto \red{m = 7} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Substituting m = 7 in equation (1)

$\rm:\longmapsto{2(7) - n = 32} \\ \\ \rm:\longmapsto{14 - n = 34} \: \: \: \: \\ \\ \rm:\longmapsto{n = 14 - 34} \: \: \: \: \\ \\ \rm:\longmapsto \red{n = - 18} \: \: \: \: \: \: \: \:$

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