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

Polynomials of higher degree

Q: What is a polynomial of higher degree?

A polynomial of higher degree is one whose highest power of x is four or more, for example a quartic has degree 4 and a quintic has degree 5. Its graph is a single smooth continuous curve, and the degree sets the maximum number of x-intercepts and turning points.

Q: How do you find the end behaviour of a polynomial?

Look at the leading term, the term with the highest power of x, because it dominates for large values. The parity of the degree and the sign of the leading coefficient decide the ends: an even degree sends both ends the same way, up if the leading coefficient is positive and down if it is negative, while an odd degree sends the two ends opposite ways.

Q: What does a repeated factor do to the graph of a polynomial?

The multiplicity of a factor fixes the shape at that x-intercept. A single factor gives a straight crossing, a double factor makes the curve touch the axis and turn back like a parabola vertex, and a triple factor makes the curve flatten and pass through as a stationary point of inflection.

Q: How many turning points can a polynomial have?

A polynomial of degree n has at most n minus 1 turning points and at most n x-intercepts. For example a quartic of degree 4 has at most 3 turning points and at most 4 x-intercepts, but it may have fewer of each.

Q: How do you sketch a quartic from its factored form?

Use the even degree and the sign of the leading coefficient to fix the end behaviour, set each factor to zero for the x-intercepts, apply the multiplicity rule at each one, mark the y-intercept by evaluating at x equals 0, then join everything with a smooth curve having at most three turning points.

Q: What is the difference between a double and a triple factor?

A double factor has even multiplicity, so the curve touches the x-axis and turns back without crossing. A triple factor has odd multiplicity, so the curve crosses the axis, but it flattens as it does so, forming a stationary point of inflection.

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Theory

A polynomial of degree \(n\), such as a quartic \(y=ax^{4}+\dots\), is sketched from its factored form: each real zero gives an \(x\)-intercept, and the multiplicity of a factor fixes the shape there — a single factor crosses, a double factor touches, and a triple factor flattens through. The end behaviour follows from the parity of the degree and the sign of the leading coefficient, and a degree-\(n\) polynomial has at most \(n\) \(x\)-intercepts and at most \(n-1\) turning points.

A polynomial is a sum of terms \(P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{1}x+a_{0}\) with whole-number powers. The degree \(n\) is the highest power and \(a_{n}\) (with \(a_{n}\neq 0\)) is the leading coefficient. Degree \(4\) is a quartic and degree \(5\) a quintic; the graph is always a single smooth, continuous curve with no breaks or corners.

The end behaviour — where the two arms of the curve point — is set by the leading term \(a_{n}x^{n}\), because it dominates for large \(|x|\). An even degree sends both ends the same way (both up if \(a_{n}>0\), both down if \(a_{n}<0\)); an odd degree sends the ends in opposite directions.

A repeated factor changes the shape at its \(x\)-intercept. The power of the factor is its multiplicity: a single factor \((x-r)\) crosses straight through, a double factor \((x-r)^{2}\) touches the axis and turns back like a parabola vertex, and a triple factor \((x-r)^{3}\) flattens and passes through as a stationary point of inflection. In total the graph has at most \(n\) \(x\)-intercepts and at most \(n-1\) turning points.

Key idea. Multiplicity decides the shape at a zero: odd multiplicity crosses the axis (a triple factor crosses but flattens first), while even multiplicity touches and turns back.

\(y=0.35(x+2)(x-1)^{2}(x-3)\): single roots cross at \(x=-2,3\); the double root touches at \(x=1\)

Even-degree end behaviour: with \(a>0\) both ends rise; with \(a<0\) both ends fall

General form — degree \(n\), leading coefficient \(a_{n}\neq 0\):

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

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

Factored form — each factor gives a zero, its power is the multiplicity (the multiplicities sum to at most \(n\)):

\[P(x)=a(x-r_{1})^{m_{1}}(x-r_{2})^{m_{2}}\cdots(x-r_{k})^{m_{k}}\]

P (x) = a {(x - r_{1})}^{m_{1}} \dots {(x - r_{k})}^{m_{k}}

End behaviour — the leading term dominates, so for large \(|x|\):

\[P(x)\approx a_{n}x^{n} \qquad \#\text{intercepts}\le n,\quad \#\text{turning points}\le n-1\]

P (x) \approx a_{n} x^{n}

Multiplicity rule. At a zero of multiplicity \(m\): \(m=1\) crosses, \(m=2\) touches and turns back, \(m=3\) flattens and crosses (a stationary point of inflection). Odd \(m\) crosses; even \(m\) touches.

Sketching a polynomial from its factored form

End behaviour. Read off the degree \(n\) and the leading coefficient. Even degree → both ends the same way; odd degree → opposite ways; the sign of the leading coefficient sets up or down.
\(x\)-intercepts. Set each factor to zero. Note the multiplicity of each zero from the power of its factor.
Shape at each zero. Apply the multiplicity rule: single → cross, double → touch, triple → flatten through.
\(y\)-intercept. Evaluate \(P(0)\) (multiply the constants) to fix where the curve meets the \(y\)-axis.
Join smoothly. Draw one continuous curve that matches the ends and each intercept shape, using at most \(n-1\) turning points.

Sign-check tip. Between two consecutive distinct real zeros the value of \(P(x)\) keeps one sign, so the curve stays entirely above or below the axis there — a quick way to catch a mis-drawn arm.

Example 1 — end behaviour and parity

Describe the end behaviour of (a) \(y=x^{4}-4x^{2}\) and (b) \(y=-x^{5}+x\).

Solution

(a) \(y=x^{4}-4x^{2}\)

Leading term \(x^{4}\): degree \(4\) is even, coefficient \(+1>0\).

degree	\(=\)	\(4\ (\text{even})\)
leading coeff.	\(=\)	\(+1\)

\(\therefore\) both ends point up (as \(x\to\pm\infty,\ y\to+\infty\))

(b) \(y=-x^{5}+x\)

Leading term \(-x^{5}\): degree \(5\) is odd, coefficient \(-1<0\).

degree	\(=\)	\(5\ (\text{odd})\)
leading coeff.	\(=\)	\(-1\)

\(\therefore\) ends oppose: as \(x\to-\infty,\ y\to+\infty\); as \(x\to+\infty,\ y\to-\infty\)

y = - x^{5} + x

Example 2 — intercepts and their shapes

For \(y=(x+3)(x-1)^{2}(x-2)\), give the degree, the \(x\)-intercepts with the shape at each, and the \(y\)-intercept.

Solution

Degree — add the multiplicities \(1+2+1\):

degree

\(=\)

\(4\), leading coeff. \(+1\)

Zeros and multiplicities — set each factor to \(0\):

\(x=-3\)	\(\to\)	single \(\Rightarrow\) cross
\(x=1\)	\(\to\)	double \(\Rightarrow\) touch
\(x=2\)	\(\to\)	single \(\Rightarrow\) cross

\(y\)-intercept — put \(x=0\):

\(P(0)\)

\(=\)

\((3)(-1)^{2}(-2)=-6\)

\(\therefore\) quartic, ends both up; crosses at \(x=-3,2\), touches at \(x=1\); \(y\)-intercept \(-6\)

y = (x + 3) {(x - 1)}^{2} (x - 2)

Example 3 — sketch with a triple factor

Sketch \(y=(x+1)(x-2)^{3}\), showing the intercepts and their shapes.

Solution

End behaviour — degree \(1+3=4\) (even), leading coeff. \(+1\):

ends

\(\to\)

both up

Zeros — apply the multiplicity rule:

\(x=-1\)	\(\to\)	single \(\Rightarrow\) cross
\(x=2\)	\(\to\)	triple \(\Rightarrow\) flatten through

\(y\)-intercept — \(P(0)=(1)(-2)^{3}=-8\).

\(\therefore\) crosses at \(x=-1\), flattens through at \(x=2\), \(y\)-intercept \(-8\)

y = (x + 1) {(x - 2)}^{3}

Example 4 — two even factors

Sketch \(y=-(x+2)^{2}(x-3)^{2}\), and state where the graph lies relative to the \(x\)-axis.

Solution

End behaviour — degree \(2+2=4\) (even), leading coeff. \(-1<0\):

ends

\(\to\)

both down

Zeros — both factors are squared, so both are double:

\(x=-2\)	\(\to\)	double \(\Rightarrow\) touch
\(x=3\)	\(\to\)	double \(\Rightarrow\) touch

Sign — write \(y=-\big[(x+2)(x-3)\big]^{2}\); the bracket is squared, so:

\(\big[(x+2)(x-3)\big]^{2}\)	\(\ge\)	\(0\)
\(y=-\big[\,\cdots\,\big]^{2}\)	\(\le\)	\(0\)

\(\therefore\) the graph lies on or below the \(x\)-axis, touching at \(x=-2\) and \(x=3\); \(y\)-intercept \(P(0)=-(2)^{2}(-3)^{2}=-36\)

y = - {(x + 2)}^{2} {(x - 3)}^{2}

Common pitfalls

A double factor touches — it does not cross. At \((x-r)^{2}\) the curve meets the axis and turns back, like a parabola vertex. Even multiplicity touches; odd multiplicity crosses.

Even degree does not mean both ends up. The direction of the ends also depends on the sign of the leading coefficient: with \(a_{n}<0\) an even-degree graph has both ends pointing down.

A triple factor still crosses. \((x-r)^{3}\) flattens at \(x=r\) as a stationary point of inflection, but the curve does pass through the axis — do not draw it as a touch.

Frequently asked questions

What is a polynomial of higher degree?

A polynomial of higher degree is one whose highest power of \(x\) is four or more — a quartic has degree \(4\) and a quintic has degree \(5\). Its graph is a single smooth, continuous curve, and the degree sets the maximum number of \(x\)-intercepts and turning points.

How do you find the end behaviour of a polynomial?

Look at the leading term, since it dominates for large \(|x|\). The parity of the degree and the sign of the leading coefficient decide the ends: an even degree sends both ends the same way (up if the leading coefficient is positive, down if negative), while an odd degree sends the two ends opposite ways.

What does a repeated factor do to the graph of a polynomial?

The multiplicity of a factor fixes the shape at that \(x\)-intercept. A single factor gives a straight crossing, a double factor makes the curve touch and turn back like a parabola vertex, and a triple factor makes it flatten and pass through as a stationary point of inflection.

How many turning points can a polynomial have?

A polynomial of degree \(n\) has at most \(n-1\) turning points and at most \(n\) \(x\)-intercepts. For example a quartic (degree \(4\)) has at most \(3\) turning points and at most \(4\) \(x\)-intercepts, but it may have fewer of each.

How do you sketch a quartic from its factored form?

Fix the end behaviour from the even degree and the sign of the leading coefficient, set each factor to zero for the \(x\)-intercepts, apply the multiplicity rule at each one, mark the \(y\)-intercept by evaluating at \(x=0\), then join everything with a smooth curve having at most three turning points.

What is the difference between a double and a triple factor?

A double factor has even multiplicity, so the curve touches the axis and turns back without crossing. A triple factor has odd multiplicity, so the curve crosses, but it flattens as it does so, forming a stationary point of inflection.

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Determining the rule for the graph of a polynomial

Polynomials of higher degree

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Sketching a polynomial from its factored form

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Frequently asked questions

Theory