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17Calculus Precalculus - Polynomial Zeroes/Roots

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As explained on the main polynomials page, zeroes are where the graph of a polynomial crosses the x-axis. They are also called x-intercepts, roots and, less common, poles. We will use the three main terms, x-intercepts, zeroes and roots, interchangeably on this page, so that you become comfortable with all of them. (The term zeroes usually refers to points on graphs while the term roots is more general and includes zeroes on the graph as well as complex numbers. We will not be picky about using these distinctions since many textbooks and instructors use them interchangebly.)

Note - This is one of the few precalculus pages where we will not assume we are talking only about real numbers. Complex numbers are also possible on this page. Click here to review complex numbers. For a complete discussion of complex zeroes, see this page.

What Are Zeroes and Roots of a Polynomial?

The zeroes of a polynomial are where the graph of the polynomial crosses the x-axis and why they are called x-intercepts. On graphs, you can also say these are points where \(y=0\), which means the same thing. These points have special meaning in many contexts.

One of the nice things about polynomials is that there are exactly the same number of roots as the highest power in the polynomial. So, if we have a polynomial \(g(x)=x^4+3x^3-1\), the highest power is 4 and so we know that we have 4 roots.
Special Notes
1. Notice that we did not say that the roots are all real values. From the context, it is natural to assume this but that is not the case. For example, the graph of the polynomial \(y=x^2+1\) never crosses the x-axis but it does have two roots, both complex.
2. Just because the roots exist, does not mean we can find them. In fact, it is sometimes not possible to find them exactly from the equation and we may have to go to our calculuator to approximate them.

Find The Roots Given The Polynomial

Finding the roots of a polynomial is pretty straightforward. The technique requires you to understand the concept of the zero product rule since it is used heavily when calculating the roots. Basically the technique requires us to
1. move all terms to one side of the equal sign, leaving zero on the other side; this may also entail setting the polynomial equal to zero;
2. factor the polynomial;
3. use the zero product rule and solve for the x-values.

In the last step, we may be able to solve for the x-values directly or we may need to approximate the values using our calculator or graphing utility. Here is a quick example.

Example

Find the roots of the polynomial \(x^2-4\).

\(x^2-4=0\)

set the polynomial equal to zero

\((x+2)(x-2)=0\)

factor

\((x+2)=0 \to x=-2\) or
\((x-2)=0 \to x=2\)

use the zero product rule and solve

\(x=-2; x=2\)

final answers

This takes some practice, so let's work some practice problems.

Practice

Solve these problems giving your answers in exact form.

Find all zeros of \(p(x)=x^3-4x^2+x-4\) given that \(+i\) is a zero.

Problem Statement

Find all zeros of \(p(x)=x^3-4x^2+x-4\) given that \(+i\) is a zero.

Solution

PatrickJMT - 1638 video solution

video by PatrickJMT

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Find the roots of \(f(x)=8x^4-36x^3-20x^2\).

Problem Statement

Find the roots of \(f(x)=8x^4-36x^3-20x^2\).

Solution

MIP4U - 1641 video solution

video by MIP4U

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Given that two zeros are \(x=2\) and \(x=3\), find the remaining zeros for \(f(x)=x^4-10x^3+37x^2-\) \(60x+36\).

Problem Statement

Given that two zeros are \(x=2\) and \(x=3\), find the remaining zeros for \(f(x)=x^4-10x^3+37x^2-\) \(60x+36\).

Solution

PatrickJMT - 1634 video solution

video by PatrickJMT

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Find all the roots of \(f(x)=x^3-12x^2+49x-78\).

Problem Statement

Find all the roots of \(f(x)=x^3-12x^2+49x-78\).

Solution

MIP4U - 1644 video solution

video by MIP4U

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Find the roots of the polynomial \(f(x)=x^3-3x^2-13x+15\).

Problem Statement

Find the roots of the polynomial \(f(x)=x^3-3x^2-13x+15\).

Solution

MIP4U - 1646 video solution

video by MIP4U

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Find the zeros of the polynomial \(f(x)=x^3-4x^2-11x+2\).

Problem Statement

Find the zeros of the polynomial \(f(x)=x^3-4x^2-11x+2\).

Solution

MIP4U - 1647 video solution

video by MIP4U

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Find the roots of \(f(x)=-x^3+16\).

Problem Statement

Find the roots of \(f(x)=-x^3+16\).

Solution

MIP4U - 1648 video solution

video by MIP4U

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Find the roots of \(f(x)=x^3-4x^2+4x-16\).

Problem Statement

Find the roots of \(f(x)=x^3-4x^2+4x-16\).

Solution

MIP4U - 1651 video solution

video by MIP4U

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Find the roots of \(p(x)=x^3+4x^2+9x+36\).

Problem Statement

Find the roots of \(p(x)=x^3+4x^2+9x+36\).

Solution

MIP4U - 1655 video solution

video by MIP4U

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Find all the real zeros of \(f(x)=x^3+x^2-10x+8\).

Problem Statement

Find all the real zeros of \(f(x)=x^3+x^2-10x+8\).

Solution

PatrickJMT - 1636 video solution

video by PatrickJMT

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Find the roots of \(f(x)=2x^5-3x^4+2x^3-3x^2-\) \(144x+216\).

Problem Statement

Find the roots of \(f(x)=2x^5-3x^4+2x^3-3x^2-\) \(144x+216\).

Solution

MIP4U - 1652 video solution

video by MIP4U

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Find all the real zeros of \(f(x)=3x^4-8x^3-37x^2+\) \(2x+40\).

Problem Statement

Find all the real zeros of \(f(x)=3x^4-8x^3-37x^2+\) \(2x+40\).

Solution

PatrickJMT - 1635 video solution

video by PatrickJMT

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Find the roots of \( f(x) = x^4 - 3x^3 - 12x^2 + \) \(54x - 40 \).

Problem Statement

Find the roots of \( f(x) = x^4 - 3x^3 - 12x^2 + \) \(54x - 40 \).

Solution

MIP4U - 1650 video solution

video by MIP4U

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Find the real zeros of \(p(x)=2x^5+x^4-2x-1\).

Problem Statement

Find the real zeros of \(p(x)=2x^5+x^4-2x-1\).

Solution

Khan Academy - 1642 video solution

video by Khan Academy

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