Zeros and Successive Differences of Polynomial Functions (College Board AP® Precalculus): Study Guide

Zeros of polynomial functions

What is a zero of a polynomial function?

• If $a$ is a number and $p\left(a\right)=0$

  • then $a$ is called a zero of the polynomial function $p$

  • Equivalently, $a$ is a root of the equation $p\left(x\right)=0$

• Zeros of polynomial functions can be real or complex (non-real) numbers

How are zeros related to factors?

• If $a$ is a real number, then $(x-a)$ is a linear factor of $p$ if and only if $a$ is a zero of $p$

  • I.e. if $(x-a)$ is a linear factor of $p$, then $p\left(a\right)=0$

  • Or if $p\left(a\right)=0$, then $(x-a)$ is a linear factor of $p$

• E.g. if $p\left(x\right)=(x-3)(x+1)(x-5)$, then the zeros of $p$ are $x=3$, $x=-1$, and $x=5$

  • This lets you write down the solutions to $p\left(x\right)=0$ if you have $p\left(x\right)$ in fully factorised form

• Conversely, if you know that $x=2$ is a zero of a polynomial, then $(x-2)$ must be a factor

What is the multiplicity of a zero?

• If a linear factor $(x-a)$ is repeated $n$ times in the factorisation of a polynomial

  • the corresponding zero has multiplicity $n$

• E.g. $p\left(x\right)=(x-1{)}^2(x+4{)}^3\left(x+7\right)$ has

  • a zero at $x=1$ with multiplicity 2

  • a zero at $x=-4$ with multiplicity 3

  • and a zero at $x=-7$ with multiplicity 1

• A polynomial function of degree $n$ has exactly $n$ complex zeros when counting multiplicities

  • A 'complex zero' here means a zero of the function that is either a real or a (non-real) complex number

• E.g. a degree 5 polynomial always has exactly 5 zeros (counting multiplicities)

  • though some may be repeated (i.e. have a multiplicity greater than 1)

  • and some may be complex (i.e. non-real)

How do zeros appear on a graph?

• If $a$ is a real zero of a polynomial function $p$, then the graph of $y=p\left(x\right)$ has an $\mathit{x}$-intercept at the point $(a, 0)$

  • The graph crosses or touches the $x$-axis at that point

• The multiplicity of a zero determines the behaviour of the graph at that $x$-intercept:

• If the zero has odd multiplicity (e.g. 1, 3, 5, ...), the graph crosses the $x$-axis at that point

  • The sign of the output values changes on either side of the zero

  • If the multiplicity is 1, the graph 'cuts straight across' the $x$-axis

  • If the multiplicity is 3, 5 or more, then the curve 'levels out' as it crosses the $x$-axis

• If the zero has even multiplicity (e.g. 2, 4, 6, ...), the graph is tangent to (touches but does not cross) the $x$-axis at that point

  • The sign of the output values is the same on both sides of the zero

  • The graph touches the axis then 'turns back'

• Graph behavior at zeros with different multiplicities can be seen here:

How can zeros help solve polynomial inequalities?

• The real zeros of a polynomial are the points where the output value is zero

• Therefore they divide the $x$-axis into intervals

  • On each interval between consecutive zeros, the polynomial is either entirely positive or entirely negative

  • This means the real zeros act as endpoints for intervals satisfying polynomial inequalities

• E.g. to solve $p\left(x\right)\ge 0$

  • find the zeros first

  • then determine the sign of $p\left(x\right)$ on each interval between zeros

How can I find the zeros of a polynomial?

• If the polynomial is given in factored form, set each factor equal to zero and solve

  • E.g. for $p\left(x\right)=(x-3)(x+1)(x-5)=0$

    • $x-3=0\Rightarrow x=3$

    • $x+1=0\Rightarrow x=-1$

    • $x-5=0\Rightarrow x=5$

  • The zeros are $x=3$, $x=-1$, and $x=5$

• If the polynomial is a quadratic, you can factor it (if possible) or use the quadratic formula

• For higher-degree polynomials that are not easily factored, you can use a graphing calculator to find the real zeros

  • Enter the polynomial into the calculator and find where the graph crosses the $x$-axis

  • Or use the calculator's "zero" or "solve" function

  • E.g. to find the zeros of $g\left(x\right)=-0.167x^3+x^2-1.834$

    • you would enter the function into your calculator

    • and find that the zeros are approximately $x=-1.233$, $x=1.578$, and $x=5.643$

What about complex (non-real) zeros?

• Some zeros of a polynomial may be complex numbers

  • I.e. numbers involving $i=\sqrt{-1}$

• These do not correspond to $x$-intercepts on the graph

  • since only real zeros appear on the real number line (i.e. on the $x$-axis)

• Complex zeros always come in conjugate pairs

  • I.e. if $a+bi$ is a zero of a polynomial $p$ (where $b\ne 0$)

    • then $a-bi$ is also a zero of $p$

  • E.g. if $2+3i$ is a zero, then $2-3i$ must also be a zero

• This means non-real zeros always account for an even number (0, 2, 4, 6...) of the total zeros

Degree and successive differences

How can successive differences reveal the degree of a polynomial?

• The degree of a polynomial function can be found by examining the successive differences of the output values over equally spaced input values

• To do this:

  • Start with output values at equally spaced inputs

  • Calculate the 1st differences (differences between consecutive outputs)

    • If the 1st differences are constant, the polynomial has degree 1 (linear)

  • If not, calculate the 2nd differences (differences between consecutive 1st differences)

    • If the 2nd differences are constant, the polynomial has degree 2 (quadratic)

  • If not, continue taking successive differences until you reach a constant set of differences

    • The degree of the polynomial is equal to the least value $n$ for which the $n$th differences are constant

• E.g. consider these values at equally spaced inputs for a polynomial function $p$:

• Calculate the differences

  • 1st differences: $3, 9, 21, 39$ — not constant

  • 2nd differences: $6, 12, 18$ — not constant

  • 3rd differences: $6, 6$ — constant

• The 3rd differences are constant, so $p$ is a degree 3 (cubic) polynomial