Key takeaway: Polynomial long division divides one polynomial by another using the same repeat-and-subtract routine as numeric long division, producing a quotient polynomial and a remainder of lower degree.
Grade 11 – Algebra 2A-APR.D.6Expressions and Equations
Definition of Polynomial Long Division
Polynomial long division works like the division you learned for whole numbers. Divide the leading terms, multiply the result back across the divisor, subtract, and bring down the next term. Repeat until the leftover has a lower degree than the divisor.
The outcome is a quotient and a remainder. If the remainder is 0, the divisor is a factor of the dividend. Before starting, write both polynomials in descending order of degree and insert a zero coefficient for any missing power, so the columns stay lined up.
Formula and notation
Polynomial long division produces this statement for dividend \(P\), divisor \(D\), quotient \(Q\) and remainder \(R\):
\[ \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}, \qquad P(x) = D(x)\,Q(x) + R(x) \]
The degree of \(R\) is less than the degree of \(D\).
Key facts
- Each cycle is divide, multiply, subtract, bring down.
- Put in a 0 coefficient for every missing power before you start.
- A remainder of 0 means the divisor is a factor of the dividend.
- Check by multiplying the divisor and quotient, then adding the remainder.
Polynomial Long Division example
Divide \(x^3 + 2x^2 – 9x + 5\) by \(x^2 – x + 2\).
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Divide leading terms: \(x^3 : x^2 = x\). Multiply: \(x(x^2 – x + 2) = x^3 – x^2 + 2x\). Subtract: \(3x^2 – 11x + 5\).
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Divide again: \(3x^2 : x^2 = 3\). Multiply: \(3(x^2 – x + 2) = 3x^2 – 3x + 6\). Subtract: \(-8x – 1\).
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Degree 1 is below degree 2, so stop. Quotient \(x + 3\), remainder \(-8x – 1\).
\(\dfrac{x^3 + 2x^2 – 9x + 5}{x^2 – x + 2} = \) \(x + 3 + \dfrac{-8x – 1}{x^2 – x + 2}\).
Common mistake
Watch out
During polynomial long division, Lia subtracts \(x^3 – x^2 + 2x\) from \(x^3 + 2x^2 – 9x\), Lia gets \(x^2 – 7x\) instead of \(3x^2 – 11x\).
Why it is wrong
She subtracted only the first sign. The subtraction applies to every term of the bracket: \(2x^2 – (-x^2) = 3x^2\) and \(-9x – 2x = -11x\). Changing all the signs of the line being subtracted, then adding, avoids the slip.
Related terms and lessons
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