Polynomials & Algebraic Identities Practice

What a polynomial actually is

A polynomial is an expression built from whole-number powers of x, added together: 3x² − 5x + 2 is one, and so is x⁵. Each piece is a term, the number in front of a term is its coefficient, and the largest power is the degree. Working with polynomials comes down to two moves: expanding brackets, then collecting terms that match.

Terms match only when the power is identical. 2x² and 3x² make 5x², but 2x² and 3x stay apart, because one counts squares and the other counts x. Expansion is repeated distribution: each term in the first bracket reaches each term in the second, so two brackets of two terms produce four products before anything is collected. A polynomial can have one term, like 4x³, or several, like x² − 5x + 2. A term with no x is a constant. The degree belongs to the highest power, not to the number of terms, so x⁵ + 2x + 1 has degree 5.

What this ladder covers

A single bracket opens the ladder, and substituting into the result closes it.

• What counts: recognise a polynomial by its whole-number powers, and name the terms, coefficients and degree.
• One bracket: multiply each term inside by the factor outside, as in 3(x + 4).
• Two brackets: write out all four products for (x + 2)(x + 5) before collecting.
• A squared bracket: expand (x − 3)² as a product of two brackets rather than squaring each part.
• Collecting: add the coefficients of like terms and leave everything else alone.
• Common factors: take the largest shared factor out of every term, and evaluate the expression at a given x.

Worked examples

Expand 3(x + 4). Multiply both terms inside by 3: 3 × x = 3x and 3 × 4 = 12, so the answer is 3x + 12. The factor reaches every term, not only the first one.

Expand (x + 2)(x + 5). Four products: x × x = x², x × 5 = 5x, 2 × x = 2x, and 2 × 5 = 10. Collect the two x terms: x² + 5x + 2x + 10 = x² + 7x + 10.

Expand (x − 3)². Write the bracket twice: (x − 3)(x − 3). The products are x², −3x, −3x and 9, which collect to x² − 6x + 9. The middle term is negative and does not disappear.

Collect 2x² + 3x − x² + 5x. The x² terms are 2x² − x² = x², and the x terms are 3x + 5x = 8x. Together they give x² + 8x. Nothing joins across the two groups, because x² and x are different powers.

Factorise 6x + 9. The largest number dividing both terms is 3: 6x = 3 × 2x and 9 = 3 × 3, so the answer is 3(2x + 3). Multiplying it back out returns the expression you started with, which is the check that costs five seconds.

Evaluate p(x) = x² − 3x + 1 at x = 2. Substitute: 2² − 3 × 2 + 1 = 4 − 6 + 1 = −1. Substitution is the quickest way to test whether two expressions really are the same.

Mistakes worth watching for

The term that disappears most often is the middle one.

• (x + 2)(x + 5) = x² + 10. Only the first and last products were written. The two cross products, 5x and 2x, belong in the answer, which is why it ends as x² + 7x + 10.
• (x − 3)² = x² − 9. Squaring a bracket gives three terms, not two, because the bracket is multiplied by itself. The missing −6x is the whole difference.
• 3x + 2x² = 5x². The powers differ, so the terms cannot combine. Adding them would turn two separate sizes into one that never existed.
• 3(2x + 3) = 6x + 3. The 3 reached the first term only. Distribution has to reach every term inside the bracket, so the constant becomes 9.
• Collecting across a minus sign. In 5x − 2x the answer is 3x, but 5x − 2 written as 3x quietly drops the −2. Keep unlike terms where they are.
• Calling every expression a polynomial. x⁻¹ is not a polynomial because its power is negative, and 1/x is the same expression. The definition allows whole-number powers of x, including zero, but not negative powers or roots of x.

How the ladder works

Six rungs sit on this ladder, and the question count climbs from five in the Warm-Up to eight or nine at Mastery. You need 80% of first attempts correct to move on, all rungs are available immediately, and each retry is reshuffled.

Keep going

Build a sheet of expansions with the polynomials worksheet generator, let the general calculator handle the arithmetic while you check the algebra, or continue to quadratic equations, where a factorised polynomial is set equal to zero.

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