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What number should be subtracted from $4x^3+5x+3$ so that the resulting polynomial leaves remainder $-80$ when divided by $2x+5$?.

Let the required number to be subtracted be $K$.

let: $$P(x)=4x^3+5x+3-k$$

$$g(x)=2x+5=2(x+5/2)$$

Comparing $g(x)$ with $x-a $ , $a =\frac {-5}{2}$

Solving then gives $K=8$

but the answer in my book is $-152$. how?

4 Answers 4

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Let us define $f(x) = 4x^3 +5x +3 -k$, where $k$ is the number that needs to be subtracted

By Remainder Theorem, $f(a)$ provides the remainder when $f(x)$ is divided by $(x-a)$

$$f(-\frac{5}{2}) = -72-k = -80$$ $$\therefore k = -72+80=8$$

I agree with you that the answer should be 8.

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Let $f (x)=2x+5$ and $p (x)=4x^3+5x-3$. We have that: $$p (-\frac {5}{2}) -k = -80$$ $$\Rightarrow 4 (-\frac {5}{2})^3 +5 (-\frac {5}{2}) +3 - k = -80$$ $$-75+3-k = -80$$ $$\boxed {k = 8} $$ Hope it helps.

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    But the question mentions $-80$, so shouldn't the value of $k$ indeed be $8$? – 2017-01-28
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    @GoodDeeds We have $$(2x+5)q (x)+80 = 4x^3+5x+3-k $$ where $q (x) $ is the quotient. – 2017-01-28
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$$4x^3+5x+3=2x^2\underbrace{(2x+5)}-5x\underbrace{(2x+5)}+15\underbrace{(2x+5)}+3-75$$

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    @ ## lab bhattacharjee, what should I do then? – 2017-01-28
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    @occasionaluser, So, what is the remainder when divided by $2x+5$ – 2017-01-28
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    @ ## lab bhattacharjee, its $-72$? – 2017-01-28
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    @occasionaluser, Right! And u need $-80$ as the remainder? – 2017-01-28
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    @## lab bhattacharjee, what would be the answer, either $8$ or $-152$? – 2017-01-28
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    @occasionaluser, You already have $3-75=?$ Now you need to subtract something more to reach $-80$ right? – 2017-01-28
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(2x+5) is a factor.

So we have 2x+5=0

x=$\frac{-5}{2}$

When we put this into polynomial we get k = 8.

So your answer is correct.