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What are polynomial factors?

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The factors of any number are simply the numbers that multiply together to equal that number. For instance, 4 is a factor of 24, since 4 × 6 = 24.

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For polynomials, factors are typically other polynomials! A polynomial could look like *x*^{2} - *x* - 6, which has two factors.

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The two factors are (*x* - 3) and (*x* + 2). We can verify this by multiplying these two factors together, using the FOIL method.

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The FOIL method is a mnemonic device to remember how to multiply binomials together. It stands for

- First
- Outer
- Inner
- Last

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To multiply (*x* - 3)(*x* + 2), multiply the first of each binomial to find *x* × *x *= *x*^{2} .

- Multiply the outer terms to find that
*x*× 2 = 2*x*. - Multiply the inner terms to find that -3 ×
*x*= -3*x*. - Finally, multiply the last terms to find that -3 × 2 = -6.

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Combine the middle two terms and add all of this together to find the polynomial *x*^{2} - *x* - 6.

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Factoring by itself is a very useful skill, but this lesson will focus more on what we can use the polynomial factors to do.

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Most importantly, polynomial factors tell us the x-intercepts, zeros, or roots of our polynomial. All three of these names mean the same thing: where our polynomial crosses the x-axis.

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In the previous example we found that *x*^{2} - *x* - 6 = (*x* - 3)(*x* + 2). Since we are interested in the graph of this polynomial, we can set it equal to and see the following graph.

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Notice that this quadratic graph crosses the x-axis at (-2, 0) and (3, 0).

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This seems a bit odd as the polynomial factors (*x* - 3) and (*x* + 2). It feels a bit like these x-intercepts should be -3 and positive 2.

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What’s happening here is that since we are considering the graph *y* = (*x*-3)(*x*+3), we are trying to determine where *y* = 0. Plugging this into the equation, we see that 0 = (*x*-3)(*x*+3).

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This means that one of two things is true. Either *x - *3 = 0 or *x* + 2 = 0. Take a moment and consider why that is.

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Notice that since the left hand side of the equation is 0, we know one of the two factors must be 0 on the right hand side.

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Solve both of these equations for *x* to find that *x* = 3 or *x* = -2, which is exactly what we discovered from the graph alone!

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This is why polynomial factors are so powerful. They immediately show us where the zeroes of our polynomial are.

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If I told you that some polynomial *p*(*x*) had a factor of 4*x* + 3, you wouldn’t be able to tell me much about the polynomial, but we would know that when 4*x* + 3 = 0 we could find that *x *= -3/4 and know that this is one of the -intercepts of our graph!

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The other major use of polynomial factors is determining the behavior of our graph near the x-axis.

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For example, the graph of *y* = (*x - *2)^{2}(*x *- 2)^{3}(*x *+ 1) looks something like this:

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Notice that all of the *x*-intercepts are -1, 1 and 2, just like we would predict.

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However, we can be more specific with how the graph is behaving at each of these points.

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The *x*-intercept at -1 is the most straightforward. Since this corresponds to the polynomial factor (*x* + 1) , which is not raised to any power, the graph simply crosses the x-axis at *x* = -1.

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The x-intercept at 2 is the second most straightforward. Notice that this corresponds to the polynomial factor (*x *- 2)^{2}. Since this polynomial factor is squared, the graph acts like a parabola and only touches our *x*-axis, not crossing it!

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This is what will happen for every polynomial factor that is squared.

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Finally, we can consider the x-intercept at 1. This corresponds to the polynomial factor (*x *- 1)^{3} and it crosses the x-axis, but it behaves a bit strangely. Notice that the graph initially looks like it will not cross, but then it does cross the x-axis.

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This is because this is how the graph of *y* = *x*^{3 }behaves as well! When you see this behavior, it is due to a cubed polynomial factor.

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Ultimately, polynomial factors are a powerful and versatile tool that allow you to understand the zeros of polynomials and their graphs in much greater detail.

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