Algebraic Division – A-Level Maths Key Concepts

Question-and-answer flashcards summarising the key definitions, theorems, and procedures for algebraic division at A-Level Maths.

What is the degree of a polynomial?

The highest power of x in the polynomial.

In the polynomial 4x^5 + 6x^2 − 3x − 1, what is the degree?

5

In algebraic division, what is the divisor?

The expression you are dividing by (e.g., in (x^2 + 4x − 3)/(x + 2), the divisor is x + 2).

In algebraic division, what is the quotient?

The result of the division excluding the remainder.

In algebraic division, what is the remainder?

The constant left after division; for A-Level Maths it will always be a constant.

State the Factor Theorem.

If f(x) is a polynomial and f(a) = 0, then (x – a) is a factor of f(x).

How does the Factor Theorem help with factorising a polynomial?

Knowing a root gives you a linear factor; divide by this factor to reduce the polynomial’s degree.

Use the Factor Theorem to show that (2x + 1) is a factor of 2x^3 − 3x^2 + 4x + 3.

Set 2x + 1 = 0 → x = −1/2; f(−1/2) = 0, therefore (2x + 1) is a factor.

What is the first (subtraction) method of algebraic division?

Subtract successive multiples of the divisor to cancel the highest power of x, repeating until no terms remain.

How do you choose the multiple to subtract during algebraic division?

Match the leading term of the dividend with a term that will cancel it when multiplied by the divisor.

What is algebraic long division used for?

To divide one polynomial by another, analogous to numerical long division.

List the step-by-step process for algebraic long division.

1) Divide the leading term of the dividend by the leading term of the divisor; 2) Multiply the divisor by this result; 3) Subtract from the dividend; 4) Repeat until only a remainder remains.

What is the general form of the result after long division of polynomials?

f(x) = q(x) + r(x)/d(x), where q(x) is the quotient and r(x) the remainder.

Outline the formula (coefficient comparison) method for algebraic division.

Assume f(x) = q(x)d(x) + r(x); substitute general coefficients, evaluate at chosen x-values, and solve the simultaneous equations to find the unknown coefficients.

When is the formula method most useful?

When the divisor is linear and you need to express a polynomial in the form q(x)d(x) + r(x) or when you want to check remainders quickly.

Using the Remainder Theorem, what is the remainder when f(x) = x^3 − 3x^2 + 7x − 12 is divided by x − 3?

f(3) = 27 − 27 + 21 − 12 = 9, so the remainder is 9.

If deg f(x) = 3 and the divisor d(x) is linear, what is the degree of the quotient q(x)?

deg q(x) = deg f(x) − deg d(x) = 3 − 1 = 2.