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Year 12 Maths - Methods (Unit 3 & Unit 4) Polynomial functions

The language of polynomials

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Theory

A polynomial in standard form is \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\), a sum of whole-number powers of \(x\) with number coefficients. Its degree \(n\) is the highest power present, \(a_n x^n\) is the leading term with leading coefficient \(a_n\), and \(a_0\) is the constant term. A polynomial is monic when \(a_n=1\); you evaluate it by substituting to find \(P(a)\), and its end behaviour follows from the leading term and whether the degree is even or odd.

A polynomial is an expression built from non-negative integer powers of a variable. Written in standard form — powers in decreasing order — it is \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\), where the numbers \(a_n,a_{n-1},\dots,a_0\) are the coefficients and each \(a_kx^k\) is a term. Powers must be whole numbers, so \(x^2\) and \(x^5\) are allowed but \(\sqrt{x}=x^{1/2}\) and \(\dfrac{1}{x}=x^{-1}\) are not.

The degree is the highest power with a non-zero coefficient. That top term \(a_nx^n\) is the leading term, its coefficient \(a_n\) is the leading coefficient, and the term with no \(x\), namely \(a_0\), is the constant term. When the leading coefficient is \(1\) the polynomial is monic. Degrees \(1,2,3,4\) name the familiar linear, quadratic, cubic and quartic polynomials.

To evaluate a polynomial at \(x=a\), substitute and simplify to get the number \(P(a)\); in particular \(P(0)=a_0\), the constant term. For very large \(|x|\) the leading term \(a_nx^n\) dwarfs every other term, so the end behaviour of the graph depends only on the sign of \(a_n\) and the parity (even or odd) of the degree \(n\).

Key idea. In \(P(x)=a_nx^n+\cdots+a_0\): the degree is \(n\), the leading coefficient is \(a_n\), the constant term is \(a_0=P(0)\), and the polynomial is monic when \(a_n=1\). Always rewrite in standard form before reading these off.
End behaviour of an odd-degree polynomial with positive leading coefficientA cubic curve falling on the left and rising on the right; as x decreases without bound P(x) tends to negative infinity and as x increases without bound P(x) tends to positive infinity. x y P→−∞ P→∞
Odd degree, positive leading coefficient (e.g. \(P(x)=x^3\)): down on the left, up on the right
End behaviour of an even-degree polynomial with positive leading coefficientA quartic curve rising on both the left and the right; as x tends to plus or minus infinity P(x) tends to positive infinity. x y P→∞ P→∞
Even degree, positive leading coefficient (e.g. \(P(x)=x^4\)): both ends up

If the leading coefficient were negative, each curve would be reflected in the \(x\)-axis: the odd-degree ends become up-left and down-right, and the even-degree ends both point down.

Standard form — powers of \(x\) in decreasing order with number coefficients:

\[P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\qquad a_n\neq 0\]
P(x)=anxn+⋯+a1x+a0

Degree, leading term and leading coefficient, constant term, and the monic condition:

\[\deg P=n,\qquad \text{leading term }=a_nx^n,\qquad a_0=P(0),\qquad \text{monic}\iff a_n=1\]
degP=n,a0=P(0)

End behaviour — for large \(|x|\) the polynomial behaves like its leading term:

\[P(x)\approx a_nx^n \quad\text{as } x\to\pm\infty\]
P(x)≈anxn
End-behaviour rule. If \(n\) is even, both ends go the same way (both up when \(a_n>0\), both down when \(a_n<0\)). If \(n\) is odd, the ends go opposite ways (down-left and up-right when \(a_n>0\), the reverse when \(a_n<0\)).

How to read a polynomial's language

  1. Write it in standard form. Expand any brackets, collect like terms, and order the powers of \(x\) from highest to lowest.
  2. Read the degree and leading term. The highest power present is the degree \(n\); the term \(a_nx^n\) is the leading term and \(a_n\) is the leading coefficient. It is monic if \(a_n=1\).
  3. Identify the constant term. The term with no \(x\) is \(a_0\); it equals \(P(0)\), the \(y\)-intercept. If none is written, \(a_0=0\).
  4. Evaluate when needed. To find \(P(a)\), substitute \(x=a\) and simplify carefully with signs and powers.
  5. State the end behaviour. Use the sign of \(a_n\) and whether \(n\) is even or odd to decide which way each end of the graph goes.
Shortcut for end behaviour. Ignore every term except the leading one: sketch \(y=a_nx^n\) mentally. Even power gives a \(\cup\)/\(\cap\) pair of ends; odd power gives one end up and one down, flipped by the sign of \(a_n\).
Example 1 — naming the parts
For \(P(x)=3x^4-2x^2+7x-5\), state the degree, leading term, leading coefficient and constant term. Is it monic?
Solution
It is already in standard form — read each feature off the top power down:
degree\(=\)\(4\)
leading term\(=\)\(3x^4\)
leading coefficient\(=\)\(3\)
constant term\(=\)\(-5\)
The leading coefficient is \(3\), not \(1\), so it is not monic.
\(\therefore\) degree \(4\), leading term \(3x^4\), leading coefficient \(3\), constant term \(-5\); not monic
3x4-2x2+7x-5
Example 2 — evaluating \(P(a)\)
For \(P(x)=x^3-4x+1\), find \(P(2)\) and \(P(-1)\).
Solution
\(P(2)\)
Substitute \(x=2\):
\(P(2)\)\(=\)\((2)^3-4(2)+1\)
\(=\)\(8-8+1\)
\(=\)\(1\)
\(\therefore\) \(P(2)=1\)
\(P(-1)\)
Substitute \(x=-1\) — mind the signs:
\(P(-1)\)\(=\)\((-1)^3-4(-1)+1\)
\(=\)\(-1+4+1\)
\(=\)\(4\)
\(\therefore\) \(P(-1)=4\)
P(2)=1,P(−1)=4
Example 3 — standard form from a jumble
Write \(P(x)=2x^2+3x^3-x^2+5-x^3\) in standard form, then state its degree, leading coefficient and constant term.
Solution
Collect like terms — group the \(x^3\) and \(x^2\) terms:
\(P(x)\)\(=\)\((3x^3-x^3)+(2x^2-x^2)+5\)
\(=\)\(2x^3+x^2+5\)
Now read off the features from the standard form:
degree\(=\)\(3\)
leading coefficient\(=\)\(2\)
constant term\(=\)\(5\)
\(\therefore\) \(P(x)=2x^3+x^2+5\); degree \(3\), leading coefficient \(2\), constant term \(5\)
2x3+x2+5
Example 4 — end behaviour from the leading term
Describe the end behaviour of \(P(x)=-2x^5+3x^2-1\) as \(x\to\pm\infty\), and match it to a graph.
Solution
Identify the leading term — only it matters for large \(|x|\):
leading term\(=\)\(-2x^5\)
degree \(n\)\(=\)\(5\ (\text{odd})\)
\(a_n\)\(=\)\(-2\ (<0)\)
Odd degree with \(a_n<0\): the ends go opposite ways, up on the left and down on the right.
as \(x\to-\infty\)\(:\)\(P(x)\to+\infty\)
as \(x\to+\infty\)\(:\)\(P(x)\to-\infty\)
\(\therefore\) up-left, down-right: \(P(x)\to+\infty\) as \(x\to-\infty\) and \(P(x)\to-\infty\) as \(x\to+\infty\)
End behaviour of an odd-degree polynomial with negative leading coefficientA cubic curve rising on the left and falling on the right; as x tends to negative infinity P(x) tends to positive infinity and as x tends to positive infinity P(x) tends to negative infinity. x y P→∞ P→−∞
-2x5+3x2-1

Common pitfalls

Read the degree only after standard form. In \(2x^2+3x^3-x^2+5-x^3\) the highest power written first is not the leading term; collect like terms to \(2x^3+x^2+5\), then the degree is \(3\).
The leading coefficient is not always the first number you see. In \(7x+3x^4-5\) the leading coefficient is \(3\) (the coefficient of \(x^4\)), not \(7\). Reorder into decreasing powers first.
A missing constant means \(a_0=0\), not "no constant". The constant term of \(x^3+x\) is \(0\), so its graph passes through the origin; \(P(0)=0\).

Frequently asked questions

What is the degree of a polynomial?

The degree is the highest power of \(x\) with a non-zero coefficient, once the polynomial is in standard, expanded form. For \(3x^4-2x^2+7x-5\) the degree is \(4\); the degree controls the maximum number of turning points and the end behaviour.

What are the leading term and leading coefficient?

The leading term is the term of highest degree and the leading coefficient is the number in front of it. In \(3x^4-2x^2+7x-5\) they are \(3x^4\) and \(3\). Reorder into standard form first, since the leading term is not necessarily written first.

What does it mean for a polynomial to be monic?

A polynomial is monic when its leading coefficient is \(1\), so it looks like \(x^n+\cdots\). For example \(x^3-4x+1\) is monic, but \(2x^3-4x+1\) is not because its leading coefficient is \(2\).

How do you evaluate P(a)?

Substitute \(x=a\) and simplify. For \(P(x)=x^3-4x+1\), \(P(2)=8-8+1=1\) and \(P(-1)=-1+4+1=4\). The value \(P(0)\) always equals the constant term.

How do you find the end behaviour of a polynomial?

For large \(|x|\) the leading term dominates, so the end behaviour depends only on the sign of the leading coefficient and the parity of the degree. Even degree sends both ends the same way; odd degree sends the ends opposite ways.

What is the constant term of a polynomial?

The constant term is \(a_0\), the term with no \(x\), and it equals \(P(0)\), the \(y\)-intercept. If none is written, the constant term is \(0\); for example \(x^3+x\) has constant term \(0\).

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