The zeros of a function are the input values which make the function equal to zero. This means a is a zero of f\left(x\right) if f\left(a\right) = 0. We also refer to these solutions as roots of the equation f\left(x\right)=0. These zeros or roots appear on the graph of the function as x-intercepts.

Real zeros can also serve as endpoints for intervals that satisfy polynomial inequalities, as they represent points where the polynomial function changes its sign. This concept helps in finding intervals where the polynomial function is greater than or less than a certain value.

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The fundamental theorem of algebra says the number of complex roots of any polynomial is equal to the degree of the polynomial. Remember that complex roots refer to real and imaginary roots. The real zeros of a function will be the x-intercepts of its graph.

Recall the factor theorem says if x=a is a root of f\left(x\right) = 0, then \left(x-a\right) is a factor of f\left(x\right).

The multiplicity of a zero is the number of times that its corresponding factor appears in the function. The multiplicities of the zeros in the function will sum to the degree of the polynomial by the fundamental theorem of algebra. Zeros with different multiplicities look different graphically.

Zeros of multiplicity 1

Zeros of multiplicity 2

Zeros of multiplicity 3

Graph of y=(x+3)(x+1)^2(x-2)^3

A root of multiplicity 1 crosses through the x-axis with no point of inflection. A root with an odd multiplicity greater than 1 crosses through the x-axis with a point of inflection. Roots with even multiplicity are tangent to the axis which means they touch the x-axis, then change direction and do not cross the x-axis.

When the coefficients of a polynomial meet certain criteria, complex roots and irrational roots will come in conjugate pairs.

Complex conjugate roots theorem

If P\left(x\right) is a polynomial with real coefficients and a+bi is a root of P\left(x\right), then its complex conjugate a-bi is also a root.

Irrational conjugate roots theorem

If P\left(x\right) is a polynomial with rational coefficients and a+\sqrt{b} is an irrational root of P\left(x\right), then its conjugate a-\sqrt{b} is also a root.

Examples

Example 1

A polynomial f\left(x\right)=x^5+7x^4+17x^3+47x^2+72x-144 has zeros at x=-4 and x=-3i.

Determine the remaining zeros of f\left(x\right).

Worked Solution

Create a strategy

By the fundamental theorem of algebra, this function has 5 complex roots since the degree is 5. We were given 2 of the roots which means 3 still remain.

Since this polynomial has real coefficients, we can apply the complex conjugates theorem. We can then create factors for the known roots and use polynomial division to factor f\left(x\right) and find the remaining roots of the function.

Apply the idea

If a number k is a zero of f\left(x\right), then f\left(k\right)=0. It follows that x-k is a factor by the factor theorem. So, we have \left(x+4\right) and \left(x+3i\right) as factors of f\left(x\right).

Since this polynomial has real coefficients, we can apply the complex conjugates theorem which says if -3i is a root of the polynomial, then 3i must also be a root of the polynomial.

Multiplying together the factors we know, we get

\displaystyle \left(x+4\right)\left(x+3i\right)\left(x-3i\right)	\displaystyle =	\displaystyle \left(x+4\right)\left(x^2+9\right)	Difference of squares identity
	\displaystyle =	\displaystyle x^3+4x^2+9x+36	Distributive property

Next, we can use polynomial division to find P\left(x\right)\div \left(x^3+4x^2+9x+36\right) and factor the result to find the remaining zeros.

A figure showing the polynomial long division for x to the fifth power plus 7 x to the fourth power plus 17 x cubed plus 47 x squared plus 72 x minus 144 divided by x cubed plus 4 x squared plus 9 x plus 36. Speak to your teacher for more information.

The result, x^2+3x-4, can be factored further: x^2+3x-4=\left(x+4\right)\left(x-1\right) Setting these factors to zero and solving, we find the remaining roots are x=-4 and x=1, and the root we found earlier at x=3i.

Reflect and check

The zeros given to use were x=-4 and x=-3i, and the zeros we found were x=3i, x=-4, and x=1. We can check whether the zeros of f\left(x\right) are correct by multiplying their corresponding factors.

\displaystyle f\left(x\right)	\displaystyle =	\displaystyle \left(x-1\right)\left(x+4\right)\left(x+4\right)\left(x+3i\right)\left(x-3i\right)	Function expressed by its factors
	\displaystyle =	\displaystyle \left(x-1\right)\left(x+4\right)^2\left(x^2+9\right)	Difference of squares identity
	\displaystyle =	\displaystyle \left(x-1\right)\left(x^2+8a+16\right)\left(x^2+9\right)	Perfect square trinomial identity
	\displaystyle =	\displaystyle \left(x^2+8a+16\right)\left(x^3-x^2+9x-9\right)	Distribute \left(x-1\right)\left(x^2+9\right)
	\displaystyle =	\displaystyle x^5+7x^4+17x^3+47x^2+72x-144	Distributive property

This shows that the zeros we obtained are correct.

State the multiplicities of the zeros.

Worked Solution

Create a strategy

The full factorization of the polynomial is f\left(x\right)=\left(x-1\right)\left(x+4\right)\left(x+4\right)\left(x+3i\right)\left(x-3i\right) which can be simplified to f\left(x\right)=\left(x-1\right)\left(x+4\right)^2\left(x+3i\right)\left(x-3i\right). To determine the multiplicities of each zero, we determine the number of times that the corresponding factor of each zero occurs in the function.

Apply the idea

From the factored form of the polynomial, we see

x=1 is a zero with multiplicity 1
x=-4 is a zero with multiplicity 2
x=-3i is a zero with multiplicity 1
x=3i is a zero with multiplicity 1

Reflect and check

There are 5 roots which confirms the number of roots expected from the fundamental theorem of algebra.

Example 2

Consider the polynomial function: h\left(x\right)=\left(x-3\right)\left(x^2-4x+5\right).

State the zeros of h\left(x\right).

Worked Solution

Create a strategy

We can use the zero product property and set each factor equal to 0. This will give us one linear equation and one quadratic equation to solve. For the quadratic equation we will need to use the quadratic formula.

Apply the idea

Start by setting each factor equal to 0.

\left(x-3\right)=0 and \left(x^2-4x+5\right)=0

Solving the linear equation x-3=0 we get x=3.

To solve x^2-4x+5=0 we will use the quadratic formula.

First identify a, b, \text{ and } c. a=1, b=-4, \text{ and } c=5. Then substitute into the quadratic formula and evaluate.

\displaystyle x	\displaystyle =	\displaystyle \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}	Quadratic formula
\displaystyle x	\displaystyle =	\displaystyle \dfrac{-\left(-4\right) \pm \sqrt{\left(-4\right)^2-4\left(1\right)\left(5\right)}}{2\left(1\right)}	Substitute values for a,b, and c
\displaystyle x	\displaystyle =	\displaystyle \dfrac{4 \pm \sqrt{16-20}}{2}	Evaluate exponents and multiplication
\displaystyle x	\displaystyle =	\displaystyle \dfrac{4 \pm \sqrt{-4}}{2}	Evaluate subtraction
\displaystyle x	\displaystyle =	\displaystyle \dfrac{4 \pm 2i}{2}	Evaluate square root (recall \sqrt{-1}=i)
\displaystyle x	\displaystyle =	\displaystyle 2 \pm i	Evaluate division

So the zeros of h\left(x\right) are x=3, x=2+i and x=2-i.

Reflect and check

Notice how the complex solution came in a conjugate pair.

Write an equation for h\left(x\right) using linear factors.

Worked Solution

Create a strategy

In part (a) we found x=3, x=2+i and x=2-i. We can apply the zero product property in reverse to rewrite solutions as factors.

Apply the idea

Starting with x=3 subtract 3 from both sides so the equation is equal to 0 giving x-3=0.

For x=2+i subtract 2 and i from both sides giving x-2-i=0.

Similarly for x=2-i subtract 2 and add i to both sides giving x-2+i=0.

Finally use these equations to write the factors of h\left(x\right).

h\left(x\right)=\left(x-3\right)\left(x-2-i\right)\left(x-2+i\right)

Example 3

Given the polynomial function: p\left(x\right)=\left(x+3\right)^2\left(x-4\right)\left(x+5\right), without graphing, describe the behavior of the graph near the zero x=3.

Worked Solution

Create a strategy

Consider the multiplicity of the zero in question.

Apply the idea

x=3 is a zero with a multiplicity of 2 (even multiplicity). This means that the function will touch the x-axis at x=3 and then turn and continue in the other direction (meaning it will not cross the x-axis at this point).

Reflect and check

We can confirm this by looking at a graph of the function.

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Idea summary

The multiplicity of a zero is the number of times that its corresponding factor appears in the function.

Even multiplicity: function touches the x-axis at that zero then changes direction

Odd multiplicity: function crosses the x-axis at that zero

Complex roots come in conjugate pairs of the form a+bi and a-bi.

Degrees of polynomials

The degree of a polynomial function is the highest power of the independent variable in its expression. To determine the degree of a polynomial function, we can examine the successive differences of output values over equal-interval input values. The degree is equal to the least value n for which the successive nth differences are constant.

Consider the function represented by the following table:

an x y table, next to the table are 3 lists of numbers the first is labeled first difference, the second is labeled second difference, the third is labeled third difference

Notice the first differences, which are found by subtracting consecutive y-values in the table, are all different numbers.

Similarly the second differences, which are found by subtracting the consecutive first differences, are also different numbers.

But finally the third differences, which are found by subtracting the consecutive second differences, are constant (or all the same number).

Because the 3rd differences are the first to be constant the degree of this function is 3.

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Visualizing this function on a graph we can see that it has the familiar shape of a cubic function.

Examples

Example 4

A polynomial function, w\left(x\right), is shown in the table:

x	0	1	2	3	4	5
w(x)	1	6	31	100	229	426

Worked Solution

Create a strategy

Find the successive differences until you find the ones that are constant.

Apply the idea

1st differences: 6-1=5, 31-6=25, 100-31=69, 229-100=129, 426-229=197.

2nd differences: 25-5=20, 69-25=44, 129-69=60, 197-129=68.

3rd differences: 44-20=24, 60-44=16, 68-60=8.

4th differences: 16-24=-8, 8-16=-8.

The 4th differences are all constant \left(-1\right) so the polynomial is a 4th degree (quartic) polynomial.

Idea summary

Find successive differences by subtracting consecutive y-values. The degree of a polynomial is found by identifying which successive difference is constant.

Outcomes

1.5.A

Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.

1.5.B

Determine if a polynomial function is even or odd.

1.5A Polynomial functions and complex zeros

Introduction

Ideas

Zeros of polynomial functions

Examples

Example 1

Example 2

Example 3

Degrees of polynomials

Examples

Example 4

Outcomes

1.5.A

1.5.B

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