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I don't understand how I should go about solving the following question:

Find the remainder when polynomial $f(x)=3x^{2019}+5x^{1019}-7x+4$ is divided by $x^2-1$.

I tried to use the factor theorem, but I never encountered a problem with a divisor which, in this case, is $(x+1)(x-1)$, so I simply found $f$ of both roots, so $f(1)$ and $f(-1)$.

Allegedly the remainder is a linear polynomial in the form $ax+b$ but I fail to see how they derived that fact. Apparently they made use of simultaneous equations, but I'm not sure how or why.

Any help would be appreciated!

thekerbal
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  • The remainder must always be at least one degree less than the divisor. If remainder r(x)=$px^2+qx+b$, then you can write it as $p(x^2-1)$+$(p+q)x+b$ – insipidintegrator Jun 05 '22 at 09:44
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    Hint : Write $f(x)=q(x)(x^2-1)+r(x)$, and evaluate this equation for $x=-1$ and $x=1$. You should get $r(1)$ and $r(-1)$, and since $r$ has degree one, you should be able to conclude. – TheSilverDoe Jun 05 '22 at 09:44

2 Answers2

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Degree of the remainder less then degree of quotient then remainder is linear , $R(x)=ax+b$

$f(x)=3x^{2019}+5x^{1019}-7x+4=Q(x)(x-1)(x+1)+ax+b$

$f(1)=a+b=5$

$f(-1)=-a+b=3$

then $a=1,b=4$

$R(x)=x+4$

OR

To find remainder when $f(x)=3x^{2019}+5x^{1019}-7x+4$ is divided by $x^2-1$. By using remainder theorem

$x^2-1=0$

then

$x^2=1$

$f(x)=3x^{2018+1}+5x^{1018+1}-7x+4=3((x^2))^{1009}\times x +5((x^2))^{509}\times x -7x+4=3\times 1 \times x +5\times x\times 1\times x-7x+4=x+4$

Hence

$R(x)=x+4$

Lion Heart
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It's always the case that for polynomials $p, q$ the remainder of $$\frac{p(x)}{q(x)}$$ (for $p \neq 0$) has degree $< \deg p$.

Hint You're on the right track. By definition $$f(x) = (x^2 - 1) q(x) + r(x)$$ for some linear function $r$, and you've already computed $f(1) = r(1)$ and $f(-1) = r(-1)$. So, the linear function $r$ is the unique one whose graph passes through $(-1, r(-1))$ and $(1, r(1))$.

Travis Willse
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