Lesson 3: Dividing Polynomials

Property

To divide a polynomial by a binomial, use a process similar to long division of numbers.

1. Write the division problem with the dividend in standard form, using placeholders (e.g., $0x^2$) for any missing terms.
2. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
3. Multiply the new quotient term by the entire divisor and subtract it from the dividend.
4. Bring down the next term and repeat the process until finished.

Examples

• Find the quotient $(y^2 + 8y + 15) \div (y + 3)$. Long division shows that $y$ goes into $y^2$ $y$ times. Multiply $y(y+3)=y^2+3y$, subtract to get $5y$, bring down 15. $y$ goes into $5y$ 5 times. The result is $y+5$.
• Find the quotient $(x^3 - 2x^2 - 13x + 6) \div (x - 4)$. Using long division, the quotient is $x^2 + 2x - 5$ with a remainder of $-14$. The answer is written as $x^2 + 2x - 5 - \frac{14}{x-4}$.
• Find the quotient $(a^3 - 27) \div (a - 3)$. Use placeholders: $(a^3 + 0a^2 + 0a - 27) \div (a - 3)$. The result is $a^2 + 3a + 9$.

Explanation

This method mirrors the long division you learned for numbers. The key is to focus on dividing the leading terms at each step. Using placeholders for missing powers, like $0x^2$, is crucial to keep all the terms aligned correctly.

Property

Synthetic division is a shorthand method for polynomial division. Synthetic division only works when the divisor is of the form $x - c$.
To perform synthetic division, write the coefficients of the dividend and the value $c$ from the divisor. Bring down the first coefficient, multiply by $c$, add to the next coefficient, and repeat. The last number in the result is the remainder.

Examples

• To divide $x^3 + 4x^2 - 5x - 14$ by $x - 2$, use $c=2$ in synthetic division. The coefficients are $1, 4, -5, -14$. The resulting coefficients are $1, 6, 7$ with a remainder of $0$. The quotient is $x^2 + 6x + 7$.
• To divide $3x^3 + 8x^2 + 5x - 7$ by $x + 2$, use $c=-2$. The coefficients are $3, 8, 5, -7$. The process yields a quotient of $3x^2 + 2x + 1$ and a remainder of $-9$.
• To divide $y^4 - 8y^2 + 16$ by $y + 2$, use $c=-2$ and a placeholder for the $y^3$ and $y$ terms (coefficients $1, 0, -8, 0, 16$). The quotient is $y^3 - 2y^2 - 4y + 8$ with a remainder of $0$.

Explanation

Synthetic division is a fast-track version of long division that gets rid of the variables. It's a cleaner and quicker process, but remember its major limitation: it only works when you are dividing by a simple binomial like $(x-5)$ or $(x+2)$.

Property

When polynomial long division does not result in zero, the final leftover polynomial is the remainder. The final answer is written as the quotient plus a fraction where the remainder is the numerator and the divisor is the denominator.

Quotient + $\frac{\text{Remainder}}{\text{Divisor}}$

Examples

• Dividing $(x^2 + 5x + 7)$ by $(x+2)$ leaves a remainder of $1$, so the answer is written as the quotient plus a fraction: $x+3 + \frac{1}{x+2}$.
• When dividing $(2y^2 - 3y + 5)$ by $(y-1)$, the quotient is $2y-1$ and the remainder is $4$. This is written as $2y-1 + \frac{4}{y-1}$.
• For $(a^3 - a^2 + a + 5) \div (a+1)$, the quotient is $a^2-2a+3$ with a remainder of $2$, so the full answer is $a^2-2a+3 + \frac{2}{a+1}$.