Polynomial Division

What is polynomial division?

Polynomial division is the process of dividing two polynomials
- This is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator
To do this we use an algorithm similar to that used for division of integers
To divide the polynomial $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$ by the polynomial $D (x) = b_{k} x^{k} + b_{k - 1} x^{k - 1} + . . . + b_{1} x + b_{0}$ where k ≤ n
- STEP 1
  Divide the leading term of the polynomial P(x) by the leading term of the divisor D(x) : $\frac{a_{n} x^{n}}{b_{b} x^{k}} = q_{m} x^{m}$
- STEP 2
  Multiply the divisor by this term: $D (x) \times q_{m} x^{m}$
- STEP 3
  Subtract this from the original polynomial P(x) to cancel out the leading term: $R (x) = P (x) - D (x) \times q_{m} x^{m}$
- Repeat steps 1 – 3 using the new polynomial R(x) in place of P(x) until the subtraction results in an expression for R(x) with degree less than the divisor
  - The quotient Q(x) is the sum of the terms you multiplied the divisor by: $Q (x) = q_{m} x^{m} + q_{m - 1} x^{m - 1} + . . . + q_{1} x + q_{0}$
  - The remainder R(x) is the polynomial after the final subtraction

Division by linear functions

If P(x) has degree n and is divided by a linear function (ax + b) then

$\frac{P (x)}{a x + b} = Q (x) + \frac{R}{a x + b}$ where

ax + b is the divisor (degree 1)
Q(x) is the quotient (degree n – 1)
R is the remainder (degree 0)

Note that $P (x) = Q (x) \times (a x + b) + R$

Division by quadratic functions

If P(x) has degree n and is divided by a quadratic function (ax² + bx + c) then

$\frac{P (x)}{a x^{2} + b x + c} = Q (x) + \frac{e x + f}{a x^{2} + b x + c}$ where

ax² + bx + c is the divisor (degree 2)
Q(x) is the quotient (degree n – 2)
ex + f is the remainder (degree less than 2)

The remainder will be linear (degree 1) if e ≠ 0, and constant (degree 0) if e = 0
Note that $P (x) = Q (x) \times (a x^{2} + b x + c) + e x + f$

Division by polynomials of degree k ≤ n

If P(x) has degree n and is divided by a polynomial D(x) with degree k ≤ n
- $\frac{P (x)}{D (x)} = Q (x) + \frac{R (x)}{D (x)}$ where
  - D(x) is the divisor (degree k)
  - Q(x) is the quotient (degree n – k)
  - R(x) is the remainder (degree less than k)
- Note that $P (x) = Q (x) \times D (x) + R (x)$

Are there other methods for dividing polynomials?

Synthetic division is a faster and shorter way of setting out a division when dividing by a linear term of the form

To divide $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$ by $(x - c)$ :

Set $b_{n} = a_{n}$
Calculate $b_{n - 1} = a_{n - 1} + c \times b_{n}$
Continue this iterative process $b_{i - 1} = a_{i - 1} + c \times a_{i}$
The quotient is $Q (x) = b_{n} x^{n - 1} + b_{n - 1} x^{n - 2} + . . . + b_{2} x + b_{1}$ and the remainder is $r = b_{0}$

You can also find quotients and remainders by comparing coefficients

Given a polynomial $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$
And a divisor $D (x) = d_{k} x^{k} + d_{k - 1} x^{k - 1} + . . . + d_{1} x + d_{0}$
Write $Q (x) = q_{n - k} x^{n - k} + . . . + q_{1} x + q_{0}$ and $R (x) = r_{k - 1} x^{k - 1} + . . . + r_{1} x + r_{0}$
Write $P (x) = Q (x) D (x) + R (x)$

Expand the right-hand side
Equate the coefficients
Solve to find the unknowns q’s & r’s

Examiner Tip

In an exam you can use whichever method to divide polynomials - just make sure your method is written clearly so that if you make a mistake you can still get a mark for your method!

Worked example

Perform the division

\frac{x^{4} + 11 x^{2} - 1}{x + 3}

. Hence write

x^{4} + 11 x^{2} - 1

in the form

Q (x) \times (x + 3) + R

2-7-2-ib-aa-hl-polynomial-division-a-we-solution-1-2

Find the quotient and remainder for

\frac{x^{4} + 4 x^{3} - x + 1}{x^{2} - 2 x}

. Hence write

x^{4} + 4 x^{3} - x + 1

in the form

Q (x) \times (x^{2} - 2 x) + R (x)

2-7-2-ib-aa-hl-polynomial-division-b-we-solution-

Polynomial Division (DP IB Maths: AA HL)

Revision Note

What is polynomial division?

Division by linear functions

Division by quadratic functions

Division by polynomials of degree k ≤ n

Are there other methods for dividing polynomials?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities