Exercises: Rewrite Rational Expressions (Polynomial Long Division)

Which rational expression is an improper rational expression (one that requires polynomial long division to rewrite)?

After performing polynomial long division of $a(x) \div b(x)$, a student writes only the quotient polynomial $q(x)$ as the answer. What is missing from the answer when the remainder is not zero?

For integer long division, $17 \div 5$ gives quotient 3 and remainder 2, so $17 = 5 \cdot 3 + 2$. Which equation shows the analogous verification check for polynomial long division when dividing $a(x)$ by $b(x)$?

Divide $(x^2 + 5x + 7) \div (x + 2)$.

Write the answer in the form $q(x) + \dfrac{r(x)}{x+2}$. Then verify: multiply $(x + 2) \cdot q(x)$ and add the remainder — confirm the result equals $x^2 + 5x + 7$.

Divide $(x^2 + 3x - 10) \div (x - 2)$.

Write your answer in the form $q(x) + \dfrac{r(x)}{x-2}$.

Divide $(2x^3 - x^2 + 3x - 4) \div (x - 1)$.

Write your answer in the form $q(x) + \dfrac{r(x)}{x-1}$.

Divide $(3x^2 - 7x + 1) \div (3x + 2)$.

Write the answer in the form $q(x) + \dfrac{r(x)}{3x+2}$.

Divide $(x^3 - 27) \div (x - 3)$.

Before dividing, rewrite the dividend in complete standard form by inserting any missing degree terms with a coefficient of 0.