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 aheadif $(x - 2)^2$ 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 confidencethat 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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