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At zero, the two expressions shown both equal four. Does this prove they are equivalent?

\[(x+2)^2\quad\text{and}\quad x^2+4\]
Year 9, Chapter 8, 2 difficulty levels, No calculatorBuild a worksheet

Check every coefficient

Equivalent polynomial forms agree for every real input. Expansion and comparison of like-power coefficients gives a general check

  1. Same expression, different form Equivalent polynomial forms have the same value for every real input. Expanding every product proves the identity
  2. Keep every coefficient Coefficients include their signs. A missing power has coefficient zero
  3. A check is not a proof One disagreement disproves equivalence. Agreement at one input alone does not prove it
  4. Name the requested form A factorised product may be equivalent, but an expansion must have its brackets multiplied out

Worked example

Expand the expression

\[(2x+1)(x-3)\]
  1. \(\displaystyle 2x^2-6x+x-3\)Form all four products
  2. \(\displaystyle -6x+x=-5x\)Collect the two linear terms
  3. \(\displaystyle 2x^2-5x-3\)The coefficients are 2, −5 and −3.
  4. \(\displaystyle 2(2)^2-5(2)-3=-5\)At an input of two, both forms give negative five. This checks that input; the expansion establishes the identity

Common mistakes

Treating one matching input as proof
✕\(\displaystyle \text{one match}\Rightarrow\text{identity}\)✓\(\displaystyle \text{compare all coefficients}\)
For example, squaring the sum of x and two differs from adding four to the square of x, although both give four at zero
Dropping a coefficient sign
✕\(\displaystyle (x-3)(x+2)=x^2+x-6\)✓\(\displaystyle (x-3)(x+2)=x^2-x-6\)
The cross-products are two times x and negative three times x, giving negative x
Ignoring a missing power
✕\(\displaystyle x^2-25:\quad b=-25\)✓\(\displaystyle x^2-25:\quad b=0\)
In standard quadratic form, b is the coefficient of x. The constant negative twenty-five is c

Difficulty levels

introCheck a distributed product and complete missing coefficient entries
standardCompare binomial and special-product expansions and their coefficient tables

Questions about this topic

Can I write the terms in another order?+

Yes, provided each signed coefficient remains attached to the same power. The practice table identifies the power for each entry

What is the coefficient of x when x is absent?+

Zero. A missing linear term does not take the value of the constant term

Can one substitution show that two expressions are different?+

Yes. A single disagreement disproves equivalence. A single agreement does not prove it

Why are there no factorisation tasks in this section?+

This section checks expansion. Common factors, special factorisations and quadratic factorisation follow in later sections

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 8: Algebraic expressions and fractions

In this chapter

Expand binomial and special products8A Expanding and special products: Check equivalent expanded formsSubstitute values into a formulaFactorise by common factors and difference of perfect squares8B Common factors and difference of squares: Verify common-factor and square factorisationsFactorise four-term expressions by groupingFactorise monic and non-monic quadratic trinomials8D Quadratic factorisation: Verify quadratic factorisationsSimplify and operate on algebraic fractionsSolve equations that contain algebraic fractions