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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.

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 here are some practice problems.

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

1638 video

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

1641 video

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

1634 video

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

1644 video

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

1646 video

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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

1647 video

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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

1648 video

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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

1651 video

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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

1655 video

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

1636 video

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

1652 video

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

1635 video

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

1650 video

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

1642 video

video by Khan Academy

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Find The Polynomial Given The Roots

From the discussion above, finding the roots is pretty straightforward as long as you can factor the polynomial. Another thing you will be asked to do is find the polynomial given the roots. This, also, is pretty straightforward if you remember just a couple of things.

1. In precalculus you will almost certainly always be working with polynomials with real coefficients. This is important since the next point follows from and is dependent on this fact.
2. For complex roots of polynomials with real coefficients, the complex roots always appear in complex conjugate pairs. For example, if one of your roots is \(2+3i\), the other root is guaranteed to be \(2-3i\).
3. Another situation to watch for is something called multiplicity. The root can appear more than once and, therefore, have a multiplicity greater than one. For example, the roots of \(x^2-6x+9=(x-3)^2\) are both \(x=3\), so we say that the roots are 3 with multiplicity 2 (since it appears twice). Multiplicity just tells us how many times the root exists.

To solve a problem where you are given the roots and you need to come up with the polynomial, you just write the factors and multiply out. Also, most instructors will want the resulting polynomial to have integer coefficients. For example, if you multiply out and get a polynomial like \((1/2)x^2+3\), multiply by 2 to get \(x^3+6\). Notice this polynomial has the same roots as the first one even though it is not the same polynomial. Of course, it depends on the problem statement, so check with your instructor to see what they require.

Before jumping into the practice problems, let's watch a quick video explaining these techniques in more detail and how the roots look on a graph. This video also includes lots of examples.

MIP4U - Zeros, Factors and Graphs of Polynomial Functions [10min-2secs]

video by MIP4U

Okay, time for you to try your hand at these practice problems.

Solve these problems giving your answers in exact form.

Find the polynomial \(f(x)\) of degree 3 with zeros \(-1, ~2, ~4\) where \(f(1)=8\).

Problem Statement

Find the polynomial \(f(x)\) of degree 3 with zeros \(-1, ~2, ~4\) where \(f(1)=8\).

Solution

1637 video

video by PatrickJMT

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Find a polynomial function with real coefficients having zeros \(4\) and \(2+3i\).

Problem Statement

Find a polynomial function with real coefficients having zeros \(4\) and \(2+3i\).

Solution

1643 video

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Find a polynomial function with real coefficients having zeros \(-6\) and \(i\).

Problem Statement

Find a polynomial function with real coefficients having zeros \(-6\) and \(i\).

Solution

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Find a polynomial with real coefficients having roots \(-2/3, 3/7, -1\).

Problem Statement

Find a polynomial with real coefficients having roots \(-2/3, 3/7, -1\).

Solution

1656 video

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Find a polynomial with real coefficients having roots \(-4, -1, 3\).

Problem Statement

Find a polynomial with real coefficients having roots \(-4, -1, 3\).

Solution

1659 video

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Find a polynomial with real coefficients of degree 3 with roots \(x=-2\) (multiplicity 2) and \(x=3\) through the point \((2,80)\).

Problem Statement

Find a polynomial with real coefficients of degree 3 with roots \(x=-2\) (multiplicity 2) and \(x=3\) through the point \((2,80)\).

Solution

1654 video

video by MIP4U

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Find the real polynomial of degree 4 having roots \(x=4\) (multiplicity 2), \(x=-3\) and \(x=0\) passing through the point \((-2,-36)\).

Problem Statement

Find the real polynomial of degree 4 having roots \(x=4\) (multiplicity 2), \(x=-3\) and \(x=0\) passing through the point \((-2,-36)\).

Solution

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Find the polynomial \(f(x)\) of degree 3 where two roots are \(x=1\) and \(x=-2\), the leading coefficient is \(-1\) and \(f(3)=48\).

Problem Statement

Find the polynomial \(f(x)\) of degree 3 where two roots are \(x=1\) and \(x=-2\), the leading coefficient is \(-1\) and \(f(3)=48\).

Solution

1639 video

video by PatrickJMT

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Write an expression for a polynomial \(f(x)\) of degree 3 with zeros \(x=2\) and \(x=-2\), leading coefficient of 1 and \(f(-4)=30\).

Problem Statement

Write an expression for a polynomial \(f(x)\) of degree 3 with zeros \(x=2\) and \(x=-2\), leading coefficient of 1 and \(f(-4)=30\).

Solution

1640 video

video by PatrickJMT

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Find a polynomial with real coefficients having roots \(x=2\) (multiplicity 2) and \(x=-1-2i\).

Problem Statement

Find a polynomial with real coefficients having roots \(x=2\) (multiplicity 2) and \(x=-1-2i\).

Solution

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Find a polynomial with real coefficients and zeros \(3, -1, 4+2i\).

Problem Statement

Find a polynomial with real coefficients and zeros \(3, -1, 4+2i\).

Solution

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Find a polynomial with real coefficients having roots \( 1/4, -2/3, -2-i\).

Problem Statement

Find a polynomial with real coefficients having roots \( 1/4, -2/3, -2-i\).

Solution

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Rational Roots Test

The Rational Roots Test is quick way to get a list of all possible rational roots to a polynomial. It is a very good place to start when trying to find roots of higher order polynomials (3 or higher). Once you have a rational root or two, you can reduce the order of the polynomial using synthetic division to find irrational and complex roots.
However, that said, we never used this technique in calculus, so as far as we are concerned, you do not need to know this for calculus. But, as usual, check with your instructor to see what they require.
If you need to know this, we recommend this video explaining this technique and showing an example.

PatrickJMT - Rational Roots Test [6min-50secs]

video by PatrickJMT

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