If-x-2-2-is-a-factor-of-the-polynomial-f-x-mx-3-x-2-x-n-find-the-values-of-m-and-n-

Question Number 57000 by pieroo last updated on 28/Mar/19

If {(x + 2)}^{2} is a factor of the polynomial

f (x) = {mx}^{3} + x^{2} + x + n, find;

the values of m and n .

Commented by maxmathsup by imad last updated on 29/Mar/19

\Rightarrow f (- 2) = f^{^{'}} (- 2) = 0 b u t f (- 2) = - 8 m + 4 - 2 + n = - 8 m + n + 2

f^{^{'}} (x) = 3 {m x}^{2} + 2 x + 1 \Rightarrow f^{^{'}} (- 2) = 12 m - 4 + 1 = 12 m - 3 \Rightarrow 12 m - 3 = 0 a n d

- 8 m + n + 2 = 0 \Rightarrow m = \frac{1}{4} a n d n = 0 \Rightarrow f (x) = \frac{1}{4} x^{3} + x^{2} + x .

Answered by Smail last updated on 28/Mar/19

f (- 2) = 8 m + 4 + 2 + n

f^{'} (x) = 3 {m x}^{2} + 2 x + 1

S i n c e (x + 2) i s a d o u b l e f a c t o r o f f

t h e (x + 2) i s a s i m p l e f a c t o r o f \frac{d f (x)}{d x}

W h i c h m e a n s f^{'} (- 2) = 0 \Leftrightarrow 12 m + 4 + 1 = 0

m = - \frac{5}{12}

f (- 2) = 8 \times (- \frac{5}{12}) + 6 + n

n = \frac{8}{3}

f (x) = \frac{- 5}{12} x^{3} + x^{2} + x + \frac{8}{3}

Commented by MJS last updated on 28/Mar/19

something went wrong

Commented by Smail last updated on 28/Mar/19

I m a d e a m i s t a k e o n t h e 5 t h l i n e

f^{'} (- 2) = 3 m {(- 2)}^{2} + 2 \times (- 2) + 1 = 0

12 m - 4 + 1 = 0 \Leftrightarrow m = \frac{1}{4}

f (- 2) = \frac{1}{4} \times {(- 2)}^{3} + 4 - 2 + n = 0

n = 0

Answered by mr W last updated on 28/Mar/19

{m x}^{3} + x^{2} + x + n = {(x + 2)}^{2} (a x + b)

{m x}^{3} + x^{2} + x + n = (x^{2} + 4 x + 4) (a x + b)

{m x}^{3} + x^{2} + x + n = {a x}^{3} + (4 a + b) x^{2} + 4 (a + b) x + 4 b

4 a + b = 4 (a + b) = 1

\Rightarrow b = 0

\Rightarrow a = \frac{1}{4}

\Rightarrow m = a = \frac{1}{4}

\Rightarrow n = 4 b = 0

\frac{1}{4} x^{3} + x^{2} + x = \frac{1}{4} {(x + 2)}^{2} x

Commented by pete last updated on 29/Mar/19

thanks prof

Answered by malwaan last updated on 29/Mar/19

b y l o n g d i v i s i o n w e h a v e

n - 8 m + 2 = 0 (1)

12 m - 3 = 0 (2)

\Rightarrow m = \frac{1}{4}; n = 0

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