Multiple Roots
When you factorise a polynomial, some factors appear more than once — like $(x-2)^3$ or $(x+1)^2$. This repetition is called multiplicity, and it completely controls whether the graph crosses or merely touches the $x$-axis. Master multiplicity and sketching polynomials becomes a matter of reading the factored form.
What is the difference between a root of $P(x) = 0$ and a factor of $P(x)$? Without looking ahead — if $(x - 2)^2$ is a factor of $P(x)$, how many times does $x = 2$ appear as a root, and what do you think this means for the graph?
The entire lesson comes down to one idea: the multiplicity of a root is the exponent of its factor, and whether that multiplicity is even or odd tells you everything about the graph's behaviour at that intercept.
For a root $\alpha$ of $P(x) = a(x-\alpha)^m \cdot Q(x)$ where $Q(\alpha) \neq 0$:
- If $m$ is odd: the sign of $P(x)$ changes across $\alpha$, so the graph crosses the $x$-axis.
- If $m$ is even: the sign of $P(x)$ does not change across $\alpha$, so the graph touches the $x$-axis and turns back.
Key facts
- The definition of multiplicity: the exponent of the corresponding factor.
- A root with multiplicity $m$ contributes $m$ to the total degree.
- Odd multiplicity roots cross the axis; even multiplicity roots touch it.
Concepts
- Why sign change (or lack of it) at a root depends on the parity of multiplicity.
- How multiplicity affects the "flatness" of the graph near the root — higher multiplicity is flatter.
- The connection between roots, factors, and the degree of a polynomial.
Skills
- State the multiplicity of each root from a factored polynomial.
- Determine whether the graph crosses or touches at each $x$-intercept.
- Find all roots and their multiplicities by factoring $P(x)$.
Given a polynomial with a root $\alpha$ of multiplicity $m$, we can write $P(x) = (x - \alpha)^m \cdot Q(x)$ where $Q(\alpha) \neq 0$. The sign of $(x - \alpha)^m$ near $x = \alpha$ determines graph behaviour:
- $m$ odd: $(x - \alpha)^m$ changes sign across $\alpha$ (negative one side, positive the other). The graph crosses the axis. The larger the odd $m$, the flatter the crossing — it looks like $y = x^3$ rather than $y = x$ at that point.
- $m$ even: $(x - \alpha)^m \geq 0$ always (a non-negative quantity). So $P(x)$ cannot change sign at $\alpha$. The graph touches the axis and returns to the same side. The curve looks like a parabola ($m = 2$) or flatter ($m = 4$) at that root.
Examples:
- $y = (x - 1)^3$: root at $x = 1$ with multiplicity 3 (odd) — crosses, with a horizontal point of inflection.
- $y = (x - 2)^2(x + 1)$: double root at $x = 2$ (even) — touches; simple root at $x = -1$ (odd) — crosses.
- $y = x^3(x+2)^2$: triple root at $x = 0$ (odd) — crosses flatly; double root at $x = -2$ (even) — touches.
Multiplicity = exponent of the factor (x - ) in fully factored form; Odd multiplicity graph crosses x-axis at that root
Pause — copy the multiplicity-behaviour rules into your book: odd multiplicity at root $\alpha$ → graph crosses $x$-axis; even multiplicity → graph touches (bounces off) the $x$-axis at $\alpha$.
Quick check: For $P(x) = (x+1)^4(x-3)$, the behaviour at $x = -1$ is:
Worked examples · 3 in a row, reveal as you go
State the multiplicity of each root of $P(x) = (x - 1)^2(x + 3)^3(x - 5)$ and the total degree.
Root $x = 5$: exponent of $(x-5)$ is $1$ → multiplicity $\mathbf{1}$
Graph: touches at $x=1$ (even); crosses flatly at $x=-3$ (odd); crosses at $x=5$ (odd)
Find all roots of $P(x) = x^4 - 5x^2 + 4$ and state their multiplicities.
$P(x) = (x-1)(x+1)(x-2)(x+2)$
Graph crosses the $x$-axis at all four intercepts.
Find all roots of $x^5 - x^3 = 0$ and state their multiplicities.
Root $x = 1$: multiplicity $\mathbf{1}$ (odd) → crosses
Root $x = -1$: multiplicity $\mathbf{1}$ (odd) → crosses
$P(x)$ is degree 5 (positive leading coefficient $x^5$), so as $x \to +\infty$, $y \to +\infty$; as $x \to -\infty$, $y \to -\infty$.
Did you get this? True or false: the graph of $y = (x-2)^2(x+1)$ touches the $x$-axis at $x = 2$ and crosses at $x = -1$.
Misconceptions to fix · the traps that cost marks
Fill the gap: For $P(x) = (x-1)^2(x+3)^3(x-5)$, the sum of all multiplicities equals . The graph touches the axis at $x = $ .
Did you get this? True or false: the graph of $y = (x-3)^3$ crosses the $x$-axis at $x = 3$.
Activities · practice with the ideas
State the multiplicity of each root and whether the graph crosses or touches the $x$-axis at each root for $P(x) = (x-2)^2(x+1)(x-4)^3$.
Find all roots and their multiplicities for $P(x) = x^4 - 8x^2 + 16$.
A degree-5 polynomial has roots $x = 1$ (multiplicity 2), $x = -3$ (multiplicity 2), and $x = 4$ (multiplicity 1). Write it in factored form with leading coefficient 1.
For the polynomial $Q(x) = x^6 - x^4$, factor completely and state the multiplicity of each root.
Explain why an even-multiplicity root causes the graph to bounce off the $x$-axis rather than cross through it.
Earlier you were asked: if $(x-2)^2$ is a factor, how many times does $x = 2$ appear as a root?
The answer is that $x = 2$ appears as a root with multiplicity 2. It contributes $2$ to the total degree, and because $2$ is even, the graph touches the axis at $x = 2$ rather than crossing through it. The factor $(x-2)^2$ is always $\geq 0$, so the polynomial cannot change sign there — the graph bounces back.
Why does even multiplicity cause the "bounce"? Because raising any factor to an even power removes the ability to change sign. $(x-2)^2 = 0$ at $x=2$, but just to either side it is positive — so the function hugs the axis from one side only.
Odd one out: Which of these roots does not cause the graph to cross the $x$-axis?
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. State the multiplicity of each root of $P(x) = (x - 1)^2(x + 3)^3(x - 5)$. (2 marks)
Q2. For $y = (x - 2)^2(x + 1)$: (a) identify the $x$-intercepts and the behaviour of the graph at each; (b) find the $y$-intercept. (3 marks)
Q3. Find all roots of $x^5 - x^3 = 0$ and state their multiplicities. (2 marks)
Comprehensive answers (click to reveal)
Activity 1: 1. $x=2$: mult 2, touches; $x=-1$: mult 1, crosses; $x=4$: mult 3, crosses (flat) · 2. $u=x^2$: $u^2-8u+16=(u-4)^2$, so $x^2=4$, $x=\pm2$, both with multiplicity 2 · 3. $P(x)=(x-1)^2(x+3)^2(x-4)$ · 4. $Q(x)=x^4(x^2-1)=x^4(x-1)(x+1)$; $x=0$ mult 4, $x=1$ mult 1, $x=-1$ mult 1 · 5. $(x-\alpha)^m$ with $m$ even is always $\geq 0$, so $P(x)$ keeps the same sign on both sides of $\alpha$ — it cannot cross the axis.
Q1 (2 marks): $x=1$ mult 2 [0.5]; $x=-3$ mult 3 [0.5]; $x=5$ mult 1 [0.5]; total degree $=6$ [0.5].
Q2 (3 marks): (a) $x$-intercepts: $x=2$ (mult 2, even — touches); $x=-1$ (mult 1, odd — crosses) [2]. (b) $y = (0-2)^2(0+1) = 4$ [1].
Q3 (2 marks): $x^5-x^3 = x^3(x-1)(x+1)$ [1]. Roots: $x=0$ mult 3, $x=1$ mult 1, $x=-1$ mult 1 [1].
Five timed questions on multiplicity and graph behaviour. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering multiplicity questions. Lighter alternative to the boss.
- Multiplicity = exponent of the corresponding factor in fully factored form.
- Odd multiplicity → graph crosses the $x$-axis at that root.
- Even multiplicity → graph touches the $x$-axis and turns around.
- Multiplicity 3 (odd): crosses with a flat horizontal point of inflection.
- Sum of all multiplicities = degree of polynomial.
Next lesson: Graphing Polynomials — combining multiplicity, end behaviour, and intercepts to sketch polynomial graphs.
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