Teach Me Maths

Expanding Brackets

How to multiply every term, collect like terms, and expand squares, three brackets and polynomial products accurately.

Multiply every term

Expanding removes brackets by multiplying. Each term outside a bracket multiplies every term inside it. With two brackets, every term in the first must multiply every term in the second.

When multiplying powers of the same base, add the indices: x^2\times x^3=x^5. Only collect terms with the same letters and powers; x^2 and x are not like terms.

Worked example 1

Expand and simplify 3x(2x-5)-2(x+4)

  1. 3x(2x-5)=6x^2-15x and -2(x+4)=-2x-8

  2. Your turn. Combine -15x-2x. What is the coefficient of x?

    -15x-2x=-17x

  3. Answer: 6x^2-17x-8

Worked example 2

Expand and simplify (2x-3)(x+5)

  1. The four products are 2x^2, 10x, -3x and (-3)(5)

  2. Your turn. Work out (-3)(5)

    The constant term is -15

  3. Answer: 2x^2+7x-15. The middle term comes from 10x-3x

Explore the four products

For positive x, the area is both (x+2)(x+3) and x^2+5x+6. The algebraic identity remains true for other real x, even when a length interpretation is unsuitable.

Squares and conjugate brackets

(a+b)^2=a^2+2ab+b^2

(a-b)^2=a^2-2ab+b^2

A square has a middle term: do not just square each separate term.

(a+b)(a-b)=a^2-b^2

Here the cross terms cancel because the signs are opposite.

Worked example 3

Expand (3x+2)^2

  1. Write (3x+2)(3x+2). Its products are 9x^2, 6x, 6x and 4

  2. Your turn. Collect the terms to give the full expansion.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    9x^2+12x+4

  3. Answer: 9x^2+12x+4. The two cross products give 12x

Worked example 4

Expand (x-2)(x+2)(x+5)

  1. Multiply the conjugate pair first: (x-2)(x+2)=x^2-4
  2. Now multiply (x^2-4)(x+5): x^3+5x^2-4x-20

  3. Your turn. Check the constant by multiplying (-2)(2)(5)

    The constant is -20, as in the expansion.

  4. Answer: x^3+5x^2-4x-20

Keep larger products organised

For a linear bracket times a quadratic, expect six products before collecting. For three brackets, expand two first and simplify before multiplying by the third. A table or separate line for each multiplier helps prevent missing terms.

Worked example 5

Expand and simplify (2x-1)(x^2+3x-4)

  1. Multiplying by 2x gives 2x^3+6x^2-8x
  2. Multiplying by -1 gives -x^2-3x+4

  3. Your turn. What is the coefficient of x after collecting -8x-3x?

    -8x-3x=-11x

  4. Answer: 2x^3+5x^2-11x+4

Worked example 6

Expand and simplify (x+1)(x-3)(2x+5)-2x(x^2-4)

  1. The first two brackets give x^2-2x-3. Multiplying this by 2x+5 gives 2x^3+x^2-16x-15
  2. The remaining subtraction is -2x(x^2-4)=-2x^3+8x. Both signs matter.

  3. Your turn. Combine the two expanded expressions.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    The cubic terms cancel, leaving x^2-8x-15

  4. Answer: x^2-8x-15. A product containing cubics can simplify to a quadratic after subtraction.

Common mistakes

  • Missing a product. Multiply every term in one bracket by every term in the other.
  • Multiplying indices in a product. Add indices for the same base: x^2x^3=x^5.
  • Squaring terms separately. Include the middle term: (a+b)^2=a^2+2ab+b^2.
  • Applying a minus sign to one term only. Change every sign when subtracting a whole bracket.
  • Combining different powers. Collect only like terms, with the same variables and powers.

Now try it: Expanding Brackets practice questions

More on this topic: Expanding Brackets worksheet with full solutions