Math101 · Independent Ontario learning library
Math101Roots 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.
