Polynomial values and coefficient parameters
To evaluate a polynomial, substitute the input into every term and keep signed inputs in brackets. To find a coefficient parameter, turn each stated polynomial value or remainder into an exact equation
\[P(a)=c\]
- Substitute in every term Replace every occurrence of the variable by the stated input \(\displaystyle P(x)=2x^2-3x+1\Rightarrow P(2)=2(2)^2-3(2)+1\)
- Keep brackets around signed inputs Use brackets so the sign is included when a power is evaluated \(\displaystyle (-3)^2=9,\qquad -3^2=-9\)
- Evaluate powers first Find each required power before multiplying by its coefficient and combining terms \(\displaystyle P(-1)=2(-1)^2-3(-1)+1=6\)
- Turn conditions into equations For an unknown coefficient, substitute the stated input and solve the resulting exact equation \(\displaystyle P(x)=x^2+px+1,\quad P(2)=11\Rightarrow4+2p+1=11\Rightarrow p=3\)
Worked example
Evaluate the polynomial at the stated input
\[P(t)=3t^3-2t^2+4t-5,\qquad P(-1)\]
- \(\displaystyle P(-1)=3(-1)^3-2(-1)^2+4(-1)-5\)Replace every t by -1 and keep brackets around the signed input
- \(\displaystyle (-1)^3=-1,\qquad (-1)^2=1\)Evaluate the powers before combining the terms
- \(\displaystyle P(-1)=3(-1)-2(1)+4(-1)-5=-3-2-4-5=-14\)Multiply each power by its coefficient and add the resulting terms
Common mistakes
Leaving out a term
✕\(\displaystyle P(-2)=2(-2)^3+4=-12\)✓\(\displaystyle P(-2)=2(-2)^3-3(-2)+4=-6\)
The input must be substituted into every non-zero term of the polynomialLosing the sign in a power
✕\(\displaystyle (-3)^2=-9\)✓\(\displaystyle (-3)^2=9\)
The brackets make the negative input part of the square, and two negative factors give a positive resultTreating a parameter as the input
✕\(\displaystyle P(x)=x^2+px+1,\quad P(2)=2^2+2^2+1=9\)✓\(\displaystyle P(x)=x^2+px+1,\quad P(2)=2^2+2p+1=5+2p\)
The input replaces x, while p remains the unknown coefficient parameterDifficulty levels
introEvaluate an integer-coefficient polynomial at an integer input
standardEvaluate at an exact rational input or find one coefficient from a stated value
challengeEvaluate an exact surd or recover two coefficients from two exact conditions
examHandle exact value, zero or remainder information and verify the resulting polynomial
Questions about this topic
What does P(a) mean?+
P(a) is the value of the polynomial when every occurrence of the variable is replaced by a
Why are brackets needed around a negative input?+
Brackets show that the negative value is the base of the power. Squaring negative two with brackets gives four; without brackets, the result is negative four
How is a coefficient parameter found from P(a)?+
Substitute a into every term, set the expression equal to the stated value, and solve the resulting equation for the parameter
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