--- title: "Polynomial Division and the Factor Theorem" description: "Divide a polynomial by (x - k), find its remainder, and use a root to split a cubic into a linear factor and a quadratic quotient." canonical: https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C source: https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Polynomial Division and the Factor Theorem Divide a polynomial by (x - k), find its remainder, and use a root to split a cubic into a linear factor and a quadratic quotient. Subject: Mathematics · Area: Algebra · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C ## Ready when they can - Divide a cubic by (x - k), state the quotient and remainder, and check by multiplication - Use f(k) = 0 to identify the factor (x - k), including when k is negative - Find integer-root candidates, divide out a confirmed factor, and factorise the quadratic quotient when possible ## Lesson: Polynomial Division and the Factor Theorem Last stop you checked allowed inputs for the Ridgeway pump rule. Here, an output of zero gives a polynomial factor. Find a root of a cubic, then divide out its factor to leave a quadratic. With whole-number coefficients and a nonzero constant term, any integer root must divide that constant term. Try both signs of its divisors. These are candidates, not confirmed roots. Some cubics need another method, and some quadratic quotients have no further integer factors. For a polynomial, division by (x - k) gives f(x) = (x - k)Q(x) + R. Q(x) is the quotient; R is a constant remainder. Substitute x = k: the product is zero, so f(k) = R. This is the remainder theorem. Thus f(k) = 0 exactly when x - k is a factor, which is the factor theorem. It works in both directions. For k = -4, the matching factor is x + 4, since -4 + 4 = 0. *(drawing: A zero result confirms the factor used in the division.)* Write terms in descending powers. Insert 0 for missing powers, as in x³ + 0x² - 4x + 1. Divide the current leading term by the divisor's leading term. Multiply that result by the whole divisor, subtract the whole product, then bring down the next term. Repeat until only a constant remains. This remainder may be negative. A cubic divided by a linear expression has a quadratic quotient: x³ / x = x², and 2x³ / x = 2x². **Example.** Divide x³ - 2x² - 5x + 6 by x - 1. First, x³ / x = x². Multiply: x²(x - 1) = x³ - x². Subtract and bring down -5x to get -x² - 5x. Next, -x² / x = -x. Multiply: -x(x - 1) = -x² + x. Subtracting gives -6x. Bring down 6 to get -6x + 6. Each subtraction removes the current leading term. **Example.** For f(x) = x³ - 2x² - 5x + 6, f(1) = 1 - 2 - 5 + 6 = 0, so x - 1 is a factor. Finish the division: -6(x - 1) = -6x + 6, leaving remainder 0 and quotient x² - x - 6. Find two numbers with product -6 and sum -1: -3 and 2. Thus f(x) = (x - 1)(x - 3)(x + 2). Set each bracket to zero to get roots 1, 3 and -2. After dividing out a confirmed factor, factorise the quotient if possible. **Example.** Let p(x) = x³ + 2x² - 3x + 4. On division by x - 2, the remainder is p(2) = 8 + 8 - 6 + 4 = 14. Long division gives quotient x² + 4x + 5. Therefore p(x) = (x - 2)(x² + 4x + 5) + 14. Since the remainder is not zero, x - 2 is not a factor. Substitution finds the remainder; division also finds the quotient. To evaluate a polynomial, replace every x with the input. Calculate powers, then products, then sums. Put negative inputs in parentheses. An unknown coefficient stays as a letter. For g(x) = x³ + ax - 12, g(2) = 8 + 2a - 12 = 2a - 4. If 2 is a root, set 2a - 4 = 0, giving a = 2. If division by x - 2 has remainder 6, set 2a - 4 = 6 instead, giving a = 5. Check a division by multiplying the divisor by the quotient and adding the remainder. Expand and compare every coefficient with the dividend. This can find a missing remainder. For example, (x - 1)(x² + x + 2) = x³ + x - 2. To obtain x³ + x + 5, add R with -2 + R = 5, so R = 7. A factorisation has remainder 0, so multiplying its factors must give the original polynomial. A repeated factor gives a repeated root, not another different root. For example, 2(x - 1)²(x + 3) has two distinct roots: 1 and -3. Keep the repeated factor in the full factorisation. Also keep the leading coefficient: this product starts with 2x³. Knowing the roots does not tell you that the leading coefficient is 1. A cubic need not have three different real roots. **Recap.** Dividing by x - k leaves remainder f(k). If f(k) = 0, x - k is a factor. Divide it out, factorise the quotient if possible, and check by multiplying back. *The lesson, practice questions and flashcards were written for this dataset.* ## Practice 32 questions on this page, each with its working shown. ## Needs first - [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc) - [Solving quadratic equations by factorising](https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU) ## Opens up - [Partial Fractions for Rational Integrands](https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c) - [The Limit Laws and Indeterminate Forms](https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx) - [Rational Functions, Asymptotes and Curve Sketching](https://lightmysky.com/learn/mathematics/rational-functions-asymptotes-and-curve-sketching-mt_U-f9YTNb2o) - [Sketching a Curve from Its Factorised Form](https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh) - [Roots of Polynomials and Roots of Unity](https://lightmysky.com/learn/mathematics/roots-of-polynomials-and-roots-of-unity-mt_XAISygqdQr)