Question

If (x+1) and (x-1) are the factors of f(x) = x^3 + 2 ax + b . Calculate the values of a and b

Answer 1

Answer:

$\huge\boxed{\fcolorbox{blue}{orange}{HELLO\:MATE}}$

Step-by-step explanation:

GIVEN:

♦A polynomial$f(x) = x^{3}+2ax+b$.

♦$(x+1) \:and\:(x-1)$ are the factors of f(x).

TO FIND:

♦The values of a and b.

ANSWER:

In first case:

$(x+1) is\: a \:facto\:r of\: f(x)$

So, $x+1=0$

=>$x =-1$

Therefore f(-1) =0.

=>$x^{3}+2ax+b =0$

=>$(-1) ^{3}+2×a×(-1) +b =0$

=>$-1-2a+b=0$

=>$b -2a=1$ ..................... (1)

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In second case:

$(x-1) \:is\:a\:factor\:of \:f(x)$

So, $(x-1) =0$

=>$x =1$

Therefore f(1)=0

=>$x^{3}+2ax+b =0$

=>$(1) ^{3}+2×a×(1) +b =0$

=>$1+2a+b=0$

=>$2a + b=-1$ ................ (2)

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Now,

$\large\red{\boxed{equ^{n}1+equ^{n}2}}$

=>$b - 2a = 1$

$± b ± 2a =±1$

___________

=>$2b =0$

=>$b =\dfrac{0}{2}$

$\large\green{\boxed{ b =0}}$

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Substituting value of b =0 in equation 1,we have;

=>$b-2a =1$

=>$0 -2a =1$

=>$-2a =1$

=>$a =\dfrac{-1}{2}$

$\large\purple{\boxed{ a =\dfrac{-1}{2}}}$

Therefore values of a and b are $\dfrac{-1}{2}$ and 0 respectively.

$\huge\orange{\boxed{a=\dfrac{-1}{2} \:and\: b=0}}$