Search topics

Search

Verify binomial and special-product expansions

Start a set of 8
1 OF 15

At x = 0, (x + 2)² and x² + 4 both equal 4. Does this prove they are equivalent?

Year 10, Chapter 10, 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

Check the proposed expansion of (2x + 1)(x − 3)

\[(2x+1)(x-3)\]
  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 x = 2 this agrees with 5 × −1. This is a useful check; 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, (x + 2)² and x² + 4 match at zero but differ elsewhere
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 2x and −3x, giving −x.
Ignoring a missing power
✕\(\displaystyle x^2-25:\quad b=-25\)✓\(\displaystyle x^2-25:\quad b=0\)
In ax² + bx + c, b is the coefficient of x. The constant −25 is c.

Difficulty levels

introCheck a distributed product and complete its coefficient table
standardCompare binomial and special-product expansions and their coefficients

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

Where is the mixed exam practice?+

Use the first topic in 10A for mixed binomial and special-product exam questions. This topic focuses on coefficient checks at intro and standard levels

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs

In this chapter

Expand binomial products and special products10A Expanding and special products: Verify binomial and special-product expansions10B Common factors and difference of squares: Factorise common factors and differences of squares10B Common factors and difference of squares: Verify common-factor and square-difference products10C Quadratic factorisation and fractions: Factorise a quadratic expression completelySimplify an algebraic fraction by factorising and cancelling10C Quadratic factorisation and fractions: Verify monic and simple non-monic factorisationsVerify quadratic fraction cancellations and exact valuesSolve a quadratic equation by taking square rootsSolve a quadratic equation using the null factor lawSet up and solve a quadratic application problemIdentify and interpret parabola features10G Parabola features and transformations: Distinguish linear and quadratic relationsRead and use circle equations and graphsRead and use exponential growth and decay graphs10H Circles and exponential graphs: Classify a relation as linear, quadratic, circle or exponentialConnect reciprocal rules, tables and graphs