Year 11 Methods (Unit 1 & 2) Polynomials

The Language Of Polynomials

Q: What is the degree of a polynomial?

The degree is the highest power of x that appears. In two x cubed minus five x squared minus x plus six, the degree is three.

Q: What is the leading coefficient?

After writing the polynomial with powers going from highest to lowest, the leading coefficient is the number in front of the highest power term. For two x cubed minus five x squared minus x plus six it is two.

Q: How do you evaluate a polynomial at a value?

Substitute the value for x and simplify, keeping brackets around negatives. For P of x equals x cubed plus two x squared minus five x minus six, P of minus two equals four.

Q: How does the leading term affect the graph?

For large size of x the leading term dominates, so it sets the end behaviour. A positive leading coefficient on a cubic falls to the left and rises to the right, and a negative one does the reverse.

20 practice questions 1 video lesson Theory + worked examples

Master the language of polynomials for Queensland Year 11 Mathematical Methods (QCAA). A polynomial is an expression built from whole-number powers of x, and this topic gives you the vocabulary to describe one precisely.

You will learn to name the degree, leading term and coefficients, decide whether an expression is a polynomial, evaluate it for a value, expand from its factors, and equate coefficients — the groundwork for polynomial work.

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Theory

In Year 11 Mathematical Methods (QCAA, Unit 1), a polynomial is an expression built from whole-number powers of \(x\), such as \(2x^3-5x^2-x+6\). This page covers the language of polynomials: the degree, the leading term and coefficients, deciding whether an expression is a polynomial, evaluating \(P(a)\), expanding from factors, and equating coefficients.

A polynomial in \(x\) is a sum of terms of the form \(a\,x^n\) where each power \(n\) is a whole number \((0,1,2,3,\dots)\). Its general form is \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\). Expressions with a negative or fractional index, or a variable in a denominator or under a root, are not polynomials.

The degree is the highest power of \(x\) that appears. The leading term is the term of highest degree, and its coefficient is the leading coefficient \(a_n\). The constant term \(a_0\) is the term with no \(x\) (it is the value of \(P(0)\)).

To evaluate \(P(a)\), substitute \(x=a\) and simplify. Two polynomials are identical when their coefficients match term by term — the basis of equating coefficients to find unknowns.

Read it from the standard form. Write the polynomial with powers descending: the highest power is the degree, the front coefficient is the leading coefficient, and the lone number is the constant term.

A cubic with a positive leading coefficient falls to the left and rises to the right.

A negative leading coefficient reflects the end behaviour: rises left, falls right.

The general polynomial of degree \(n\):

\[P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\]

P (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}

The value at \(x=a\) (substitution):

\[P(a)=a_na^n+a_{n-1}a^{n-1}+\dots+a_1a+a_0\]

P (a) = a_{n} a^{n} + \dots + a_{0}

Identical polynomials have equal coefficients of every power:

\[b_kx^k+\dots=c_kx^k+\dots\ \Rightarrow\ b_k=c_k\ \text{for all }k\]

b_{k} = c_{k}

End behaviour is set by the leading term. For large \(|x|\) the term \(a_nx^n\) dominates: a cubic with \(a_n>0\) falls to the left and rises to the right; with \(a_n<0\) it does the opposite.

How to read and use a polynomial

Standardise: rewrite with powers descending, filling any missing power with a zero coefficient.
Identify: the degree is the highest power, the leading coefficient is at the front, and the constant term is \(P(0)\).
Apply: substitute a value to evaluate \(P(a)\), expand a product of factors to standard form, or equate matching coefficients to find any unknowns.

Example 1 — Degree and coefficients

For \(P(x)=6-x+2x^3-5x^2\), state the degree, the leading coefficient, the constant term and the coefficient of \(x\).

Solution

Write in standard form (powers descending):

\(P(x)\)

\(=\)

\(2x^3-5x^2-x+6\)

The highest power is \(x^3\), so the degree is 3. The leading term is \(2x^3\), so the leading coefficient is 2. The lone number is the constant term 6. The \(x\)-term is \(-x\), so the coefficient of \(x\) is \(-1\).

Degree \(3\); leading coefficient \(2\); constant term \(6\); coefficient of \(x\) is \(-1\).

Example 2 — Is it a polynomial? Evaluate \(P(a)\)

Let \(P(x)=x^3+2x^2-5x-6\). Explain why \(P\) is a polynomial and state its degree, then evaluate \(P(-2)\).

Solution

Every power of \(x\) is a whole number \((3,2,1,0)\) with no root or denominator, so \(P\) is a polynomial of degree 3.

Evaluate — substitute \(x=-2\):

\(P(-2)\)	\(=\)	\((-2)^3+2(-2)^2-5(-2)-6\)
	\(=\)	\(-8+2(4)+10-6\)
	\(=\)	\(-8+8+10-6\)
	\(=\)	\(4\)

\(P\) is a degree \(3\) polynomial and \(P(-2)=4\).

P (- 2) = 4

Example 3 — Expand from factors

Expand \((x+3)(x-1)(2x-1)\) and state the coefficient of \(x^2\).

Solution

Multiply the first two factors:

\((x+3)(x-1)\)	\(=\)	\(x^2-x+3x-3\)
	\(=\)	\(x^2+2x-3\)

Multiply by the remaining factor \((2x-1)\):

\((x^2+2x-3)(2x-1)\)	\(=\)	\(2x^3-x^2+4x^2-2x-6x+3\)
	\(=\)	\(2x^3+3x^2-8x+3\)

\((x+3)(x-1)(2x-1)=2x^3+3x^2-8x+3\); the coefficient of \(x^2\) is \(3\).

2 x^{3} + 3 x^{2} - 8 x + 3

Example 4 — Equate coefficients

Given \((x^2+ax+b)(x-3)=x^3-4x^2+cx+6\) for all \(x\), find \(a\), \(b\) and \(c\).

Solution

Expand the left-hand side:

\((x^2+ax+b)(x-3)\)	\(=\)	\(x^3-3x^2+ax^2-3ax+bx-3b\)
	\(=\)	\(x^3+(a-3)x^2+(b-3a)x-3b\)

Match the \(x^2\) coefficients:

\(a-3\)	\(=\)	\(-4\)
\(a\)	\(=\)	\(-1\)

Match the constant terms:

\(-3b\)	\(=\)	\(6\)
\(b\)	\(=\)	\(-2\)

Match the \(x\) coefficients to get \(c\):

\(c\)	\(=\)	\(b-3a\)
	\(=\)	\(-2-3(-1)\)
	\(=\)	\(1\)

\(a=-1,\ b=-2,\ c=1\).

a = - 1, b = - 2, c = 1

Common pitfalls

Judging the degree from the written order. The degree is the highest power, not the last power written. In \(6-x+2x^3-5x^2\) the degree is \(3\), from the \(2x^3\) term.

Calling everything a polynomial. \(\sqrt{x}\), \(x^{-1}=\dfrac{1}{x}\) and \(x^{1/2}\) are not polynomial terms — powers must be whole numbers.

Sign slips when evaluating. Keep brackets around a negative value: \((-2)^2=4\) but \(-2^2=-4\). Substitute carefully, one term at a time.

Missing a term when expanding. Every term of the first bracket must multiply every term of the second. Collect like terms before reading a coefficient.

Frequently asked questions

What is the degree of a polynomial?

The degree is the highest power of \(x\) that appears. In \(2x^3-5x^2-x+6\) the degree is \(3\).

What is the leading coefficient?

After writing the polynomial with powers descending, the leading coefficient is the number in front of the highest-power term. For \(2x^3-5x^2-x+6\) it is \(2\).

How do you know if an expression is a polynomial?

Every power of the variable must be a whole number, with no variable under a root or in a denominator. So \(x^3-4x\) is a polynomial, but \(\sqrt{x}\) and \(\dfrac{1}{x}\) are not.

How do you evaluate a polynomial at a value?

Substitute the value for \(x\) and simplify, keeping brackets around negatives. For \(P(x)=x^3+2x^2-5x-6\), \(P(-2)=4\).

What does equating coefficients mean?

Two polynomials are identical only if the coefficient of every power matches. Setting matching coefficients equal gives equations you can solve for unknown constants.

How does the leading term affect the graph?

For large \(|x|\) the leading term dominates, so it sets the end behaviour: a positive leading coefficient on a cubic falls left and rises right; a negative one does the reverse.

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Division Of Polynomials

The Language Of Polynomials

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How to read and use a polynomial

Common pitfalls

Frequently asked questions

Theory