Math101 · Independent Ontario learning library
Math101learn.math101.caRoots and Zeros
A zero of a function is an input $r$ for which $f(r)=0$. For a polynomial, zero, root of $f(x)=0$, $x$-intercept (when real), and factor $x-r$ describe linked aspects of the same value.
Zeros solve polynomial equations and locate where models vanish. Multiplicity links algebraic factors to local graph behaviour and degree.
Intuition and core definition
A zero of a function is an input $r$ for which $f(r)=0$. For a polynomial, zero, root of $f(x)=0$, $x$-intercept (when real), and factor $x-r$ describe linked aspects of the same value. Multiplicity records how many times a factor repeats.
Notation, language, and conditions
The factor theorem states $f(r)=0$ exactly when $x-r$ is a factor. If $f(x)=(x-r)^m g(x)$ with $g(r)\ne0$, then $r$ has multiplicity $m$. A real graph usually crosses at roots of odd multiplicity and touches/turns at roots of even multiplicity, though local shape also depends on other factors.
Why this idea matters
Roots connect equations, factors, and graph intercepts, but multiplicity determines whether a graph crosses or only touches the axis.
A dependable method
- Set the polynomial equal to zero.
- Factor completely or use an appropriate root method.
- Apply the zero-product property to each factor.
- Record distinct roots and multiplicities.
- Substitute roots and compare with graph intercept behaviour.
Worked example
Representations and interpretation
Factored form displays roots and multiplicities; a graph displays intercept location and touch/cross behaviour; a table shows zero output. Each representation emphasizes a different feature of the same algebraic condition.
Reasoning about variations
Not every complex root is an $x$-intercept because the real coordinate plane displays only real inputs. Real-coefficient polynomials have nonreal roots in conjugate pairs.
Common mistakes
How to check your work
- Substitute each root into the original polynomial.
- Multiply factors and compare degree and coefficients.
- Ensure multiplicities sum to the expected degree after all complex roots are included.
Practice
- Find zeros of $(x-5)(x+1)^2$.
- What factor corresponds to root $r=-3$?
- How does an even-multiplicity real zero usually appear?
Answers and brief solutions
Show answers
- $x=5$ and $x=-1$ (multiplicity $2$) $x-5=0$ or $x+1=0$.
- $x+3$ $x-r=x-(-3)=x+3$.
- The graph touches the axis and turns The factor does not change sign across an even power.
Synthesis and transfer
A factored polynomial model reveals critical input values immediately; evaluating the sign on neighbouring intervals distinguishes odd-multiplicity crossings from even-multiplicity turns.
For $f(x)=(x-1)^2(x+3)$, the zeros are $1$ with multiplicity two and $-3$ with multiplicity one. Near $x=1$, the squared factor stays nonnegative, so the graph touches the axis and turns; near $x=-3$, the sign changes and the graph crosses. The total multiplicity matches the polynomial's degree three. Expanding may hide this behaviour, while factoring makes it immediate. Conversely, a graph can suggest zero locations and multiplicity parity but cannot by itself certify exact algebraic values. Substitution verifies each proposed root, and polynomial division confirms its associated factor.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find zeros of $(x-5)(x+1)^2$.
- $x-5=0$ or $x+1=0$.
End of lesson
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