[Math] Why do we automatically assume that when we divide a polynomial by a second degree polynomial the remainder is linear

algebra-precalculuspolynomialsquadratics

Given question:

If a polynomial leaves a remainder of $5$ when divided by $x − 3$ and a remainder of $−7$ when divided by $x + 1$,
what is the remainder when the polynomial is divided by $x^2 − 2x − 3$?

Solution:

We observe that when we divide by a second degree polynomial the remainder will generally be linear. Thus
the division statement becomes
$p(x) = (x^2 − 2x − 3)q(x) + ax + b $

Can someone please explain at a PRE-CALCULUS level? Thanks

Best Answer

Because by definition the quotient and the remainder of the division of a polynomial $p_1(x)$ by a polynomial $p_2(x)$ are polynomials $q(x)$ (the quotient) and $r(x)$ (the remainder) such that

  1. $p_1(x)=p_2(x)q(x)+r(x)$;
  2. $r(x)=0$ or its degree is smaller than the degree of $p_2(x)$.

In particular, if $p_2(x)$ is a quadratic polynomial, then the degree of $r(x)$ will be at most $1$.

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