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Forming Polynomial Equations with Roots

Instructor: Laurie Smith
In this lesson you'll learn how to form polynomial equations when given the roots of the equation and look at some examples. Sample problems will include those involving multiple roots and squares.

Polynomial Equations

A polynomial equation is an equation that has multiple terms made up of numbers and variables. Polynomials can have different exponents. The degree of a polynomial is its highest exponent. The degree tells us how many roots can be found in a polynomial equation. For example, if the highest exponent is 3, then the equation has three roots.

The roots of the polynomial equation are the values of x where y = 0. If we know the roots of the polynomial equation, we can use them to write the polynomial equation.

Example #1

Let's write a polynomial equation with roots 2, 3, and -1.

First, we take the roots. We know that x is equal to those values when y = 0. This means we can re-write this equation so that it equals 0, which gives us the factors of the polynomial. The factors of a polynomial are those terms that can be multiplied together to make up the polynomial. If x = 2, then (x - 2) is a factor of the polynomial.

The factors have the opposite signs of the roots.

Next, we write the factors of the polynomial together and multiply:

(x - 2)(x - 3)(x + 1)

In order to multiply binomials, which are polynomials that have two terms, we must be sure to multiply every term in the first factor by every term in the second factor. We start with two factors at a time. We use a box to help organize the multiplication:


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Notice, the two red terms are like terms, which means they have the same variables with the same exponents. Since they're like terms, we combine them, adding -2x and -3x to get -5x.

Now that we've multiplied two factors together, we'll take the result and multiply it by the third factor. Again, we'll use a box to organize the multiplication because it can get confusing when identifying what we have and haven't multiplied.


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When we multiply, we get like terms. In this case, we have two sets of like terms. The blue terms both have an x2, so we add them together. The red terms both have an x to the first power, so we add them as well. The answer is the result of this multiplication. Notice, that as we started with three roots, our answer has a degree (the largest exponent) of 3.

Example #2

Next, let's write a polynomial equation that has roots 2, 2, and -3.

This example contains a multiple root of 2 because it occurs twice. Just like in the last example, we have three roots, which means we'll end up with a polynomial that has a degree of 3.

First, we'll rewrite the roots as factors. Since the 2s are positive, they'll become negative in the factor. Since the 3 is negative, it will become positive in the factor.


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As you get more comfortable, this step can be skipped and you can jump straight to writing the factors.

(x - 2)(x - 2)(x + 3)

Now let's multiply. Again, we want to start with only two of the factors. It's usually easier to start with the two that are the same.


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Notice that we combined the like terms, and -2x plus -2x is equal to -4x.

Finally, we will want to multiply the result that we just got by the last factor of (x + 3).


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Again, we have two sets of like terms to combine. -4x2 plus 3x2 is -x2. 4x plus -12x is -8x.

Our final answer is a third degree polynomial.

Example #3

Finally, let's write a polynomial equation given the roots -1, 1, 4, and -4.

This polynomial has four roots, so what would the degree of the polynomial be? Since it has four roots, it will have a largest exponent of 4.

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