C. Polynomial and Rational Functions · Polynomial Functions and Equations

Factored Form: Zeros, Their Order, and Sketching

C1.5 C1.3About 70 minutes

Learning goals

I can read the zeros of a polynomial function and their order from its factored form.
I can describe how the graph behaves at a zero of order 1, 2 or 3.
I can use a sign chart to find where a polynomial function is positive or negative.
I can sketch a polynomial function from its factored form, and write a factored form from a graph.

Words to know

factored form
A polynomial written as a product of factors, such as f(x) = −2(x + 1)²(x − 2).

order (multiplicity) of a zero
The number of times a factor (x − r) appears. In (x − 2)³ the zero 2 has order 3.

sign chart
A table showing whether each factor, and so the whole function, is positive or negative on each interval between zeros.

point of inflection
A point where the graph changes the way it bends. A zero of order 3 is one: the graph flattens there as it crosses.

Factors tell you where the graph meets the x-axis

Remember from Grade 10 and 11: if y = a(x − r)(x − s), the zeros are r and s. The same idea works for any degree: if f(x) = a(x − r₁)(x − r₂)...(x − rₙ), then f(rₖ) = 0 because one factor becomes zero. So each factor (x − r) gives a zero x = r, an x-intercept of the graph.

Factors can repeat. In f(x) = (x + 1)²(x − 2), the factor (x + 1) appears twice, so x = −1 is a zero of order 2 (also called multiplicity 2). The order changes what the graph does at that zero.

Near a zero, a factor of order 1 behaves like a line, order 2 like a parabola, and order 3 like y = x³. Odd orders cross the x-axis; even orders touch it and turn back.

Order of the zero	What the graph does there	Does the sign of f(x) change?
1	crosses the x-axis (looks like a line)	yes
2	touches the x-axis and turns back (looks like a parabola)	no
3	crosses, flattening out as it passes (looks like y = x³)	yes
even (2, 4, ...)	touches and turns back	no
odd (1, 3, ...)	crosses	yes

Behaviour at a zero depends on its order.

A routine for sketching

Degree and leading coefficient (add the exponents; multiply the leading coefficients). This gives the end behaviour.
Zeros and their orders. Mark them on the x-axis.
y-intercept: substitute x = 0.
Signs. Start from the right end (you know its direction from step 1) and move left. At each zero of odd order the sign switches; at a zero of even order it does not. A sign chart does the same job factor by factor.
Draw a smooth curve through the intercepts with the right end behaviour and the right behaviour at each zero.

A sketch shows the shape and the intercepts. The exact heights of the turning points need technology or calculus, so you do not have to find them.

Worked example

Example 1: Sketching a cubic with three simple zeros

Sketch f(x) = (x + 2)(x − 1)(x − 3).

Degree 3, leading coefficient 1 > 0: the graph rises from Q3 to Q1.
Zeros: x = −2, 1, 3, each of order 1, so the graph crosses at each.
y-intercept: f(0) = (2)(−1)(−3) = 6.
Signs: the sign chart below gives negative for x < −2, positive for −2 < x < 1, negative for 1 < x < 3, positive for x > 3.
Draw: up from Q3, cross at −2, over a hill through (0, 6), down through 1, a valley, up through 3 and on to Q1.

A cubic with zeros −2, 1, 3, y-intercept 6, and end behaviour Q3 to Q1.

Sign chart for f(x) = (x + 2)(x − 1)(x − 3). Each factor switches sign at its own zero; multiplying the signs in each column gives the sign of f(x) on that interval.

Worked example

Example 2: A repeated zero

Sketch g(x) = −2(x + 1)²(x − 2).

Degree 2 + 1 = 3, leading coefficient −2 × 1 × 1 = −2 < 0: rises on the left, falls on the right (Q2 to Q4).
Zeros: x = −1 (order 2: touches and turns back) and x = 2 (order 1: crosses).
y-intercept: g(0) = −2(1)²(−2) = 4.
Signs from the right: for x > 2 the graph is negative (the right end falls). Crossing at 2 switches the sign, so g(x) > 0 for −1 < x < 2. At −1 the order is even, so the sign stays positive for x < −1.
Draw: down from Q2 to touch the x-axis at −1, back up over a hill through (0, 4), down through 2 and on to Q4.

See the graph below.

g(x) = −2(x + 1)²(x − 2) touches the x-axis at the double zero −1 and crosses at the single zero 2. The y-intercept is 4.

Worked example

Example 3: A triple zero

Sketch h(x) = (x − 1)³(x + 2).

Degree 4, leading coefficient 1: both ends rise (Q2 to Q1).
Zeros: x = 1 (order 3: crosses with a flattened S shape) and x = −2 (order 1: crosses).
y-intercept: h(0) = (−1)³(2) = −2.
Signs: positive for x > 1; negative for −2 < x < 1 (switched at the odd zero 1); positive for x < −2 (switched again at −2).
Draw: down from Q2, cross at −2 into a valley below the x-axis through (0, −2), then rise and flatten as it crosses at 1, then rise to Q1.

A quartic shaped like a tilted U with a flat crossing at x = 1.

Worked example

Example 4: From a graph to an equation

A cubic graph touches the x-axis at x = −3, crosses at x = 2, and passes through (0, −9). Find its equation in factored form.

Touching at −3 means order 2: factor (x + 3)². Crossing at 2: factor (x − 2). Degree 3 is used up.
So f(x) = a(x + 3)²(x − 2) for some a.
Use (0, −9): a(9)(−2) = −9, so −18a = −9 and a = 1/2.

f(x) = ½(x + 3)²(x − 2)

Just for fun

A zero of order 2 walks up to the x-axis, says hello, and goes back the way it came. A zero of order 3 walks right through, but slows down to admire the view first.

Hands-on activity

The sign walk

You need: masking tape or a long piece of string for a number line on the floor (or a hallway), and sticky notes or scraps of paper.

Make a number line from −4 to 4 on the floor. Choose a function such as f(x) = (x + 3)(x + 1)²(x − 2).
Put a note at each zero showing its order: −3 (1), −1 (2), 2 (1).
Stand at the far right end. The degree is 4 and a = 1, so you start above the axis: hold your arms up.
Walk left. At each zero, switch your arms (up ↔ down) if the order is odd; keep them where they are if the order is even.
Say the sign of f(x) on each interval as you walk. Check by substituting one test value in each interval.

What to look for: positive for x > 2, negative for −1 < x < 2, still negative for −3 < x < −1 (no switch at the double zero), positive for x < −3. Test: f(0) = (3)(1)(−2) = −6, negative ✓.

Your turn

State the zeros, their orders and the degree of f(x) = (x − 4)²(x + 1)³.

At which zeros does g(x) = x(x − 3)² cross the x-axis, and where does it only touch?

Give the end behaviour and y-intercept of f(x) = −(x + 2)(x − 1)(x − 4).

On which intervals is f(x) = (x + 1)(x − 2)² positive, and where is it negative?

Describe a sketch of f(x) = (x + 3)(x + 1)(x − 2).

Write a factored form: cubic, a = 1, crosses at x = −2 and touches at x = 3.

For f(x) = −(x + 1)³(x − 2), give the end behaviour and y-intercept.

More practice: warm-up

What zero does the factor (x − 5) give?

What is the order of the zero −2 in (x + 2)⁴? Does the graph cross there?

Does a graph cross the x-axis at a zero of order 2?

Find the y-intercept of f(x) = (x − 1)(x + 2)(x + 3).

State the zeros of f(x) = 2x(x + 5).

More practice: on your own

For f(x) = (2x − 1)(x + 3)², give the zeros and their behaviour, the end behaviour and the y-intercept.

Make a sign chart for f(x) = −x(x − 2)(x + 3)².

A quartic with a > 0 touches the x-axis at −2, crosses at 0 and at 3, and passes through (1, −9). Find its equation.

Explain why f(x) = (x − 1)²(x + 1)² is never negative.

Factor f(x) = x³ − 4x² + 4x and describe its graph.

Challenge

A graph comes down from Q2, touches the x-axis at −2, crosses at 1, leaves through Q4, and passes through (0, 8). Find a polynomial of least degree for it.

For f(x) = (x + 4)²(x + 1)(x − 3)⁴(x − 5)³, find the degree, the end behaviour and every interval where f(x) < 0.

Explain, using the factor (x − 1)³, why the graph of y = (x − 1)³ flattens out at x = 1 instead of crossing steeply.

Check your understanding

Quiz

At which zero does f(x) = (x + 3)²(x − 1) only touch the x-axis?

What is the y-intercept of f(x) = −(x − 2)(x + 1)(x + 3)?

Which equation matches a cubic that crosses at x = 4, touches at x = −1, and falls to the right?

On which interval is f(x) = (x + 2)(x − 1)(x − 3) negative?

How does the graph of y = (x − 2)³(x + 5) behave at x = 2?

A cubic touches the x-axis at x = 1, crosses at x = −2, and passes through (0, 4). What is its equation?

For parents and teachers

Big idea (C1.5): a factor (x − r) repeated n times gives a zero of order n at x = r. Odd order: the graph crosses; even order: it touches and turns. Together with the end behaviour and the y-intercept, this is enough to sketch the graph.

How to help: Ask your teen to explain a sketch out loud: "Which way does the right end go? What happens at each zero? Where is the y-intercept?" If they can say it, they can draw it. Check sketches with a free graphing tool such as Desmos.

Common difficulties:

Sign errors with zeros: (x + 3) gives x = −3, not 3.
Forgetting to multiply the leading coefficients of factors like (2 − x) or (3x + 1).
Switching the sign at a double zero.
Trying to find exact turning points; a sketch only needs the shape and the intercepts.