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
\[a\left(b+c\right)=ab+ac\]
- Distribute through every term Multiply the outside factor by each term inside the bracket \(\displaystyle 3\left(x+4\right)=3x+12\)
- Collect like powers only Terms can be combined only when their variable parts have the same powers \(\displaystyle 2x^2+3x\ne5x^2\)
- Use the matching factor pattern Choose a factorisation pattern that reproduces the original polynomial when expanded \(\displaystyle x^2-25=\left(x-5\right)\left(x+5\right)\)
- 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
\[\left(2x-3\right)\left(x+4\right)\]
- \(\displaystyle 2x\times x=2x^2,\quad2x\times4=8x\)Multiply the first term in the first bracket by both terms in the second bracket
- \(\displaystyle -3\times x=-3x,\quad-3\times4=-12\)Multiply the second term in the first bracket by both terms in the second bracket
- \(\displaystyle 2x^2+8x-3x-12=2x^2+5x-12\)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 bracketNot 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 itCombining 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 addedDifficulty 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 12: Continuous probability distributions
In this chapter
12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables