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Simplify algebraic terms and collect like terms

Start a set of 8
Step 3 of 7

Match the rule to the operation

A coefficient is the number multiplying the pronumeral part. In a sum, keep each sign with its term. In a product or quotient, work with factors

Collect like terms

Add or subtract coefficients only when the letters and their powers match. Reordering multiplied letters does not change the term

\[5uv-2vu=3uv\]

Multiply terms

Multiply the signed coefficients. Add exponents for each matching base. Carry every other factor through

\[6r\times(-3rs)=-18r^{2}s\]

Divide terms

Divide the signed coefficients and cancel matching factors. Subtract the denominator exponent from the numerator exponent. A denominator must be nonzero

\[\begin{gathered}\frac{28p^{3}q^{2}}{-7pq}=-4p^{2}q\\\quad(p\ne0,\ q\ne0)\end{gathered}\]

Cancel a factor completely

Equal numerator and denominator exponents leave exponent zero. A nonzero base to power zero is one, so the factor can be omitted. Keep its original restriction

\[\frac{q^{2}}{q^{2}}=q^{0}=1\quad(q\ne0)\]

Keep unlike terms separate

Different letters or powers cannot be collected. Matching at one chosen value does not make two expressions equivalent

\[4w^{2}+7w\]
Year 9, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Simplifying terms and sums

Choose the rule from the operation. Keep signs attached to terms and keep all denominator restrictions after cancelling

  1. Products Multiply coefficients and add exponents for matching bases. The displayed example uses positive integer exponents \(\displaystyle 3z^{2}\times(-5z)=-15z^{3}\)
  2. Quotients Divide coefficients and cancel matching factors. Cancelling a nonzero factor completely leaves exponent zero, so its value is one. Omit that factor, but retain its nonzero restriction \(\displaystyle \begin{gathered}\frac{16a^{3}b}{-8ab}=-2a^{2}\\\quad(a\ne0,\ b\ne0)\end{gathered}\)
  3. Like terms The letters and powers must match. Their order within a product does not matter. If signed coefficients add to zero, the whole term is zero and can be omitted \(\displaystyle 7jk^{2}-4k^{2}j=3jk^{2}\)
  4. Unlike terms Keep different powers and constants in their own groups \(\displaystyle \begin{gathered}6v^{2}+2v-3v^{2}+9\\=3v^{2}+2v+9\end{gathered}\)

Common mistakes

Adding coefficients in a product
✕\(\displaystyle (-4d)\times(6de)=2d^{2}e\)✓\(\displaystyle (-4d)\times(6de)=-24d^{2}e\)
A product requires multiplication of the signed coefficients
Adding powers in a quotient
✕\(\displaystyle \frac{u^{6}}{u^{2}}=u^{8}\)✓\(\displaystyle \frac{u^{6}}{u^{2}}=u^{4}\quad(u\ne0)\)
For a nonzero base, cancel the matching factors by subtracting the denominator exponent
Losing a sign when collecting
✕\(\displaystyle \begin{gathered}5s^{2}-8s+3s^{2}+2s\\=8s^{2}+10s\end{gathered}\)✓\(\displaystyle \begin{gathered}5s^{2}-8s+3s^{2}+2s\\=8s^{2}-6s\end{gathered}\)
Reordering terms does not change their signs. Add the linear coefficients with their signs

Difficulty levels

introRecognise simple products, quotients and short collections with positive coefficients
standardUse signed coefficients, reordered factors and cancellation. Calculate the requested coefficients or powers
challengeCollect separated groups with different powers, or cancel matching factors in quotients, including higher powers and complete cancellation
examRecognise simplified results for standard and challenge structures without extra scaffolding

Questions about this topic

When are terms alike?+

Their pronumeral factors and powers match. The order of multiplied letters does not matter

Why must a denominator be nonzero?+

Division by zero is undefined. Cancelling factors does not remove the original restriction

Does a matching value prove two expressions are equivalent?+

No. Equivalent expressions agree for every allowed value. One value giving different results is enough to show that they are not equivalent

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 2: Linear equations

In this chapter

Translate between a description and an algebraic expressionEvaluate an algebraic expression by substitutionSimplify algebraic terms and collect like termsExpand brackets and collect like terms2D Two-step linear equations: Solve a linear equation2E Linear equations with brackets: Solve a linear equationSet up and solve a practical linear equationSolve and interpret a linear inequationRepresent and read inequalities on a number lineSubstitute values into a formulaSolve for an unknown or recognise a rearranged formulaSolve a pair of simultaneous linear equations (optional extension)Solve a pair of simultaneous linear equations by elimination (optional extension)Set up a pair of simultaneous equations from a description (optional extension)Solve a practical problem with simultaneous equations (optional extension)