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Expand and factor polynomial expressions

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Expand and factor polynomial expressions

An equivalent polynomial form has the same value for every allowed value of the variable. Distribute carefully when expanding and select a factor pattern that can be verified by multiplication

\[a\left(b+c\right)=ab+ac\]
  1. 1Distribute through every term Multiply the outside factor by each term inside the bracket \(\displaystyle 3\left(x+4\right)=3x+12\)
  2. 2Collect like powers only Terms can be combined only when their variable parts have the same powers \(\displaystyle 2x^2+3x\ne5x^2\)
  3. 3Use the matching factor pattern Choose a factorisation pattern that reproduces the original polynomial when expanded \(\displaystyle x^2-25=\left(x-5\right)\left(x+5\right)\)
  4. 4Restore and verify the structure Multiply the factors or expand the brackets to check that the original expression is recovered \(\displaystyle \left(x+2\right)^2=x^2+4x+4\)
Year 12, VCE Units 3 & 4, Chapter 9, 4 difficulty levels, No calculatorBuild a worksheet

Equivalent polynomial forms

An equivalent polynomial form has the same value for every allowed value of the variable. Distribute carefully when expanding and select a factor pattern that can be verified by multiplication

\[a\left(b+c\right)=ab+ac\]
  1. Distribute through every term Multiply the outside factor by each term inside the bracket \(\displaystyle 3\left(x+4\right)=3x+12\)
  2. Collect like powers only Terms can be combined only when their variable parts have the same powers \(\displaystyle 2x^2+3x\ne5x^2\)
  3. Use the matching factor pattern Choose a factorisation pattern that reproduces the original polynomial when expanded \(\displaystyle x^2-25=\left(x-5\right)\left(x+5\right)\)
  4. Restore and verify the structure Multiply the factors or expand the brackets to check that the original expression is recovered \(\displaystyle \left(x+2\right)^2=x^2+4x+4\)

Worked example

Expand and simplify the polynomial product below

\[\left(2x-3\right)\left(x+4\right)\]
  1. \(\displaystyle 2x\times x=2x^2,\quad2x\times4=8x\)Multiply the first term in the first bracket by both terms in the second bracket
  2. \(\displaystyle -3\times x=-3x,\quad-3\times4=-12\)Multiply the second term in the first bracket by both terms in the second bracket
  3. \(\displaystyle 2x^2+8x-3x-12=2x^2+5x-12\)Collect the two like x terms to obtain the equivalent polynomial form

Common mistakes

Distributing to the first term only
✕\(\displaystyle 3\left(x+4\right)=3x+4\)✓\(\displaystyle 3\left(x+4\right)=3x+12\)
The outside factor multiplies every term inside the bracket
Not distributing a negative sign
✕\(\displaystyle -\left(x-5\right)=-x-5\)✓\(\displaystyle -\left(x-5\right)=-x+5\)
The negative outside the bracket changes the sign of both terms inside it
Combining unlike powers
✕\(\displaystyle 2x^2+3x=5x^2\)✓\(\displaystyle 2x^2+3x=2x^2+3x\)
The terms have different powers of x, so their coefficients cannot be added

Difficulty levels

introThis tier expands distributive expressions and recognises common factors
standardThis tier expands products, quadratics and standard square patterns
challengeThis tier uses grouping, repeated factors and higher-degree patterns
examThis tier selects and verifies equivalent expanded or factorised forms

Questions about this topic

What makes two polynomial forms equivalent?+

They produce the same value for every value of the variable for which both expressions are defined

Which terms are like terms?+

Like terms have exactly the same variables raised to the same powers, although their coefficients may differ

How can a factorisation be checked?+

Expand the factors and compare the result with the original polynomial

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 9: Logarithmic calculus

In this chapter

Differentiate natural-logarithmic functions9A Logarithmic differentiation: Evaluate and determine function rules9A Logarithmic differentiation: Find the maximal domain of an algebraic function9A Logarithmic differentiation: Simplify and combine algebraic fractions9A Logarithmic differentiation: Expand and factor polynomial expressions9A Logarithmic differentiation: Evaluate a polynomial or recover a coefficient parameterApply logarithmic derivatives9B Applications of logarithmic derivatives: Form and analyse an inverse relation or functionFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular line9B Applications of logarithmic derivatives: Solve a linear equation9B Applications of logarithmic derivatives: Evaluate a polynomial or recover a coefficient parameter9B Applications of logarithmic derivatives: Solve a higher-degree polynomial equation or inequality9B Applications of logarithmic derivatives: Find the maximal domain of an algebraic function9B Applications of logarithmic derivatives: Evaluate and determine function rulesAnti-differentiate reciprocal and linear-reciprocal functions9C Reciprocal and linear-reciprocal antiderivatives: Simplify and combine algebraic fractions9C Reciprocal and linear-reciprocal antiderivatives: Expand and factor polynomial expressions9C Reciprocal and linear-reciprocal antiderivatives: Solve a linear equationIntegration by recognition with logarithmic components9D Recognising logarithmic derivatives: Expand and factor polynomial expressions9D Recognising logarithmic derivatives: Round to decimal places and significant figuresFind areas involving reciprocal curvesFind the area under a logarithmic curve using its inverseLogarithmic models in context9E Areas, inverse functions and logarithmic models: Form and analyse an inverse relation or function9E Areas, inverse functions and logarithmic models: Solve a linear equation9E Areas, inverse functions and logarithmic models: Solve a higher-degree polynomial equation or inequalityPerimeter and area of standard shapes9E Areas, inverse functions and logarithmic models: Round to decimal places and significant figures