Question

If x+2 is a factor of ${{x}^{2}}+mx+14$, then m =
[a] 7
[b] 2
[c] 9
[d] 14

Answer 1

If x+2 is a factor of ${{x}^{2}}+mx+14$, then m =

[a] 7

[b] 2

[c] 9

[d] 14

Hint: Recall factor theorem. According to factor theorem x-a is a factor of p(x) if p(a) = 0. Use factor theorem and hence form an equation in m. Solve form. The value of m is the value at which x+2 is a factor of p(x).

Complete step-by-step answer:

We have $p\left( x \right)={{x}^{2}}+mx+14$

Since x+2 is a factor of p(x), we have

$p\left( x \right)=\left( x+2 \right)g\left( x \right)$, where g(x) is some polynomial in x.

Put x = -2, we get

$p\left( -2 \right)=\left( -2+2 \right)g\left( -2 \right)=0\times g\left( -2 \right)=0$

Hence x = -2 is a zero of the polynomial p(x).

Now, we have $p\left( x \right)={{x}^{2}}+mx+14$

Substituting x=-2 in the expression of p(x), we get

$p\left( -2 \right)={{\left( -2 \right)}^{2}}+\left( -2 \right)m+14=4-2m+14=-2m+18$

Hence, p(-2) = -2m+18

But p(-2) = 0

Hence, we have

-2m+18=0

Adding 2m on both sides of the equation, we get

18 = 2m

Dividing by 2 on both sides, we get

9 = m

Hence m = 9.

Hence the value of m at which p(x) is divisible by x+2 is 9.

Hence option [c] is correct.

Note: Alternative solution:

We divide p(x) by x+2 using long division method

We have

Hence, we have Remainder = 14-2(m-2)

But since x+2 is a factor of p(x), we get

14-2(m-2) = 0

i.e. 18-2m = 0

Hence m = 9.