Polynomials & Algebraic Identities Worksheets

What this worksheet covers

This handout collects expansion, collection and factorising tasks for one print. Its optional key saves checking from scratch, and generating another version supplies fresh expressions at the same setting.

A handout can ask for work such as:

• expand 4(x + 5)
• expand (x + 3)(x + 7)
• multiply out (x − 4)²
• simplify 3x² + 5x − x² + 2x
• factorise 8x + 12
• evaluate p(3) for p(x) = x² − 2x + 5
• distribute 2 across (x − 6)

Choosing a level

Distribute one factor first, then work towards reversing that process. Warm-Up might ask for 4(x + 5); Practice joins two binomials, while Steady squares a binomial and keeps its cross term. Challenge switches to factorising, and Mastery mixes expansion with factorisation on the same handout. At each stage, check by multiplying the factors back out or by collecting like powers. That reverse check distinguishes a correct-looking answer from one that lost a term.

Anything from 3 to 15 items fits. Same difficulty pins the page to a single rung; mixed difficulties walks up the six from the one you pick.

Write-in shows the four products inside (x + 3)(x + 7), and collecting them is the mark. Multiple choice A–D marks in seconds, yet a lucky option will not reveal whether the middle term was ever added.

A worked example

Check a four-product expansion, then evaluate a named rule at a chosen input.

Expand (x + 3)(x + 7). Four products: x² + 7x + 3x + 21, which collects to x² + 10x + 21. Combining 7x with 3x supplies the middle term.

Evaluate p(x) = x² − 2x + 5 at x = 3. Substituting gives 9 − 6 + 5 = 8. The minus belongs to the 2x, so the 6 is taken away rather than added.

Marking with the answer key

The cross-products or the common factor are easy to overlook when checking this work.

• (x + 3)(x + 7) answered x² + 21. The two cross-products are missing from that result. Add 7x and 3x, then collect to get x² + 10x + 21.
• (x − 4)² answered x² − 16. This is a squared difference, not a difference of squares. Multiply the parenthesis by itself: (x − 4)(x − 4) gives x² − 8x + 16.
• 4(x + 5) answered 4x + 5. The 4 reached the first term and stopped. It multiplies both, so the answer is 4x + 20.
• 8x + 12 answered 2(4x + 12). A 2 came out and a 4 stayed inside, which is not fully factorised. The highest common factor is 4, so the answer is 4(2x + 3), and multiplying back checks it. The outside factor must divide every term inside the bracket.

Printing and using it

For self-marking, leave the key on its own page. To check an expansion, multiply its factors back out and compare the result with the original expression. If the terms do not return to the starting line, inspect each product before combining like powers.

Standard gives every item a line to collect the terms on, and large-print gives the same working a wider block. Compact puts a dozen short expansions on one side for a starter. Leave a spare line after a two-parenthesis product so the learner can list four products before combining the matching x terms.

For collected work, show the Name / Date / Class header. Dyslexia-friendly type lifts the 2 clear of the x beside it, which is the difference between x² and x2. Stretch sends each third item up a rung on a page held at one difficulty, and the title can be renamed after the skill: "Factorising with a common factor".

One classroom use: keep the keys in a punched wallet at the back of the room and let pupils collect one only after they have written their own attempt, then expand the key's answer beside their own to see where the two part company.

Keep going

Work the same expressions with instant marking at the polynomials practice ladder, read the polynomials lesson for the identities behind a squared bracket, or photocopy the quadratic equation worksheets, where a factorised answer turns into a solution.

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