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Recognise and classify a polynomial

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Recognise and classify a polynomial

A polynomial in a named variable is a sum of non-negative integer powers with real coefficients. Combine like powers before reading its features. In the general form below, the leading coefficient shown in the formula is nonzero. A nonzero constant has degree 0; the zero polynomial has no defined degree here

\[\begin{gathered}P(x)=a_nx^n+\cdots+a_1x+a_0\\a_n\ne0\end{gathered}\]
  1. 1Check variable exponents Every exponent of the named variable must be a non-negative integer \(\displaystyle x^0,\quad x^2,\quad x^5\text{ are allowed powers}\)
  2. 2Allow real coefficients Coefficients may be integers, fractions or other real numbers \(\displaystyle \frac{1}{2}x^2+\sqrt{2}\text{ is a polynomial in }x\)
  3. 3Read the greatest exponent Combine like powers first. For a nonzero polynomial, the degree is the greatest exponent with a nonzero coefficient. A nonzero constant has degree 0; the zero polynomial has no defined degree here \(\displaystyle 3x^4-2x+1\Rightarrow\text{degree}=4\)
  4. 4Use zero for missing powers An absent power has coefficient zero, including an absent constant term \(\displaystyle 5x^3-2x=5x^3+0x^2-2x+0\)
  5. 5Recognise a monic polynomial A nonzero polynomial is monic exactly when its leading coefficient is 1. \(\displaystyle x^3-2x+1\text{ is monic}\)
  6. 6Choose the narrowest listed coefficient set Every integer is rational and every rational number is real. When asked for the narrowest listed set containing all coefficients, use integers if all coefficients are integers, rational numbers if all are rational but some are not integers, and real numbers if an irrational coefficient occurs \(\displaystyle \mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\)
Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Polynomial classification

A polynomial in a named variable is a sum of non-negative integer powers with real coefficients. Combine like powers before reading its features. In the general form below, the leading coefficient shown in the formula is nonzero. A nonzero constant has degree 0; the zero polynomial has no defined degree here

\[\begin{gathered}P(x)=a_nx^n+\cdots+a_1x+a_0\\a_n\ne0\end{gathered}\]
  1. Check variable exponents Every exponent of the named variable must be a non-negative integer \(\displaystyle x^0,\quad x^2,\quad x^5\text{ are allowed powers}\)
  2. Allow real coefficients Coefficients may be integers, fractions or other real numbers \(\displaystyle \frac{1}{2}x^2+\sqrt{2}\text{ is a polynomial in }x\)
  3. Read the greatest exponent Combine like powers first. For a nonzero polynomial, the degree is the greatest exponent with a nonzero coefficient. A nonzero constant has degree 0; the zero polynomial has no defined degree here \(\displaystyle 3x^4-2x+1\Rightarrow\text{degree}=4\)
  4. Use zero for missing powers An absent power has coefficient zero, including an absent constant term \(\displaystyle 5x^3-2x=5x^3+0x^2-2x+0\)
  5. Recognise a monic polynomial A nonzero polynomial is monic exactly when its leading coefficient is 1. \(\displaystyle x^3-2x+1\text{ is monic}\)
  6. Choose the narrowest listed coefficient set Every integer is rational and every rational number is real. When asked for the narrowest listed set containing all coefficients, use integers if all coefficients are integers, rational numbers if all are rational but some are not integers, and real numbers if an irrational coefficient occurs \(\displaystyle \mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\)

Worked example

Determine whether the expression below is a polynomial in t, then state its degree and coefficients

\[R(t)=3t-4+\frac{1}{2}t^4\]
  1. \(\displaystyle t^1,\quad t^0,\quad t^4\text{ all have non-negative integer exponents}\)All powers of t are non-negative integers, so R(t) is a polynomial in t
  2. \(\displaystyle R(t)=\frac{1}{2}t^4+3t-4\)Arrange the terms in descending order of power before reading the features
  3. \(\displaystyle \text{degree}=4\)The greatest exponent is 4.
  4. \(\displaystyle \text{leading coefficient}=\frac{1}{2},\qquad\text{constant coefficient}=-4\)The leading coefficient belongs to t⁴, and the constant coefficient is the term without t.
  5. \(\displaystyle \text{coefficients of }t^4,t^3,t^2,t,1:\quad \frac{1}{2},0,0,3,-4\)The missing powers have coefficient zero

Common mistakes

Rejecting non-integer coefficients
✕\(\displaystyle \frac{3}{4}x^2+1\text{ is not a polynomial}\)✓\(\displaystyle \frac{3}{4}x^2+1\text{ is a polynomial}\)
A polynomial may have fractional or irrational coefficients; its variable exponents must be non-negative integers
Reading the first written term as the degree
✕\(\displaystyle 7x^2+3x^5-1\Rightarrow\text{degree}=2\)✓\(\displaystyle 7x^2+3x^5-1\Rightarrow\text{degree}=5\)
The degree is the greatest exponent after the terms are considered, regardless of their written order
Treating a missing constant as absent information
✕\(\displaystyle 4x^3-2x\Rightarrow\text{constant coefficient}=2\)✓\(\displaystyle 4x^3-2x\Rightarrow\text{constant coefficient}=0\)
A missing constant term has coefficient zero

Difficulty levels

introOrdered integer-coefficient expressions include one invalid exponent to classify
standardShuffled powers, a missing constant and rational coefficients are included
challengeSeveral plausible non-polynomials are compared with expressions containing real irrational coefficients
examClassify every displayed expression, record degree and coefficients, then choose the narrowest listed coefficient set

Questions about this topic

What makes an expression a polynomial in x?+

Every exponent of x must be a non-negative integer. The coefficients can be any real numbers

How is the degree of a polynomial found?+

Combine like powers first. For a nonzero polynomial, the degree is the greatest exponent with a nonzero coefficient. A nonzero constant has degree 0; the zero polynomial has no defined degree here

What is the constant coefficient when no constant term is shown?+

It is 0, because an absent power has coefficient zero

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 4: Cubic polynomials

In this chapter

Recognise and classify a polynomialEvaluate a polynomial or recover a coefficient parameterAdd, subtract or multiply polynomialsDivide polynomials and state quotient and remainderAnalyse a cubic graph or determine its ruleDetermine parameters in a polynomial identityApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsSolve a cubic inequalityIntersections of a cubic with a line or quadraticFormulate and interpret a cubic model