This algebra 2 polynomial division lesson explains polynomial division and the remainder theorem in one printable page, with definitions, a method and a worked example.
Polynomial division rewrites \(p(x)\) as \(d(x)q(x)+r(x)\), where the remainder has a smaller degree than the divisor. Long division works for any divisor, and synthetic division is a shortcut for \(x-c\). The Remainder Theorem says \(p(c)\) is the remainder, and the Factor Theorem says \(x-c\) is a factor exactly when \(p(c)=0\).
Algebra 2 Polynomial Division Lesson: key facts
- Standards A-APR.B.2, A-APR.D.6, and A-APR.C.4: remainders, division of polynomials, and polynomial identities.
- Division statement: \(p(x)=d(x)q(x)+r(x)\) where the remainder has smaller degree than the divisor.
- Remainder Theorem: dividing \(p(x)\) by \(x-c\) leaves remainder \(p(c)\).
- Factor Theorem: \(x-c\) is a factor of \(p(x)\) if and only if \(p(c)=0\).
- Identity: \((x^{2}+y^{2})^{2}=(x^{2}-y^{2})^{2}+(2xy)^{2}\) generates Pythagorean triples.
What is polynomial division?
Dividing a polynomial \(p(x)\) by a polynomial \(d(x)\) works like dividing whole numbers: you find a quotient and a remainder. The result is written as a division statement.
\(p(x)=d(x)\cdot q(x)+r(x)\), where the degree of the remainder \(r(x)\) is less than the degree of the divisor \(d(x)\).
How does long division work?
- Write both polynomials in descending order, using \(0\) for any missing power.
- Divide the leading term of the dividend by the leading term of the divisor and write the result in the quotient.
- Multiply the whole divisor by that quotient term and subtract the product.
- Repeat with the new line until its degree is less than the degree of the divisor.
In the example below, \(2x^{3}: x=2x^{2}\), then \(x^{2}: x=x\), then \(6x: x=6\). The final line, \(7\), is the remainder.
What is synthetic division?
Synthetic division is a shortcut for dividing by \(x-c\). Write \(c\) at the left and the coefficients of \(p(x)\) across the top. Bring down the first coefficient, multiply by \(c\), add to the next coefficient, and repeat. For \(x-2\) we use \(c=2\); for \(x+3\) we use \(c=-3\).
What are the Remainder Theorem and the Factor Theorem?
| Theorem | Statement |
|---|---|
| Remainder Theorem | The remainder when \(p(x)\) is divided by \(x-c\) is \(p(c)\). |
| Factor Theorem | \(x-c\) is a factor of \(p(x)\) exactly when \(p(c)=0\). |
In the example, \(p(2)=2(8)-3(4)+4(2)-5=7\), which matches the remainder. For \(f(x)=x^{3}-2x^{2}-5x+6\) we find \(f(1)=0\), so \(x-1\) is a factor. Dividing gives \(x^{2}-x-6=(x-3)(x+2)\), so \(f(x)=(x-1)(x-3)(x+2)\), and the graph crosses the x-axis at \(-2\), \(1\), and \(3\).
What is a polynomial identity?
An identity is true for every value of the variables. The identity \((x^{2}+y^{2})^{2}=(x^{2}-y^{2})^{2}+(2xy)^{2}\) can be proved by expanding both sides: each equals \(x^{4}+2x^{2}y^{2}+y^{4}\). It builds Pythagorean triples: \(x=2\), \(y=1\) gives \(5\), \(3\), and \(4\).
Polynomial Division and the Remainder Theorem: frequently asked questions
When can you use synthetic division?
Use it when the divisor has the form \(x-c\), a linear factor with leading coefficient 1. For a divisor such as \(x+4\), use \(c=-4\). For other divisors, use long division.
What do you do when a power is missing?
Insert a zero coefficient for each missing power. To divide \(x^{3}-8\), write it as \(x^{3}+0x^{2}+0x-8\). Skipping the zeros shifts every column and gives a wrong quotient.
What does a remainder of zero mean?
A remainder of zero means the divisor divides the polynomial evenly. By the Factor Theorem, \(x-c\) is then a factor and \(c\) is a zero of the polynomial.
How do you find the remainder without dividing?
Use the Remainder Theorem. Substitute \(c\) into the polynomial for a divisor \(x-c\). For \(p(x)=x^{2}+3x-2\) and \(x-1\), the remainder is \(p(1)=2\).
How do you prove a polynomial identity?
Expand or simplify one side until it matches the other side, using only algebra that holds for all values. Checking one number is not a proof, but it is a quick way to catch errors.
How to use this algebra 2 polynomial division lesson
Use this algebra 2 polynomial division lesson before the worksheet, then check what you learned with the quiz.
Printable PDF · Lesson
Algebra 2 Polynomial Division and the Remainder Theorem lesson
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- Polynomial Division and the Remainder Theorem worksheet (22 problems)
- Polynomial Division and the Remainder Theorem quiz (10 questions)
- Polynomial Division and the Remainder Theorem test (45 minutes)
- Previous lesson: Polynomial Functions and Their Graphs
- Next lesson: Rational Expressions and Equations
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