Fundamental Theorem of Algebra Lesson

In this lesson, we will learn about the fundamental theorem of algebra and how to write a polynomial equation. The fundamental theorem of algebra tells us that a polynomial function of degree n will have n complex solutions. This means a 1^st degree equation would have 1 solution, a 2^nd degree equation will have 2 solutions, and so on and so forth. Now, these solutions may be repeats. In this case, you will see the term multiplicity used.
We can use these facts to write a polynomial equation given certain conditions.
Suppose our polynomial function has a degree of 3, and has the following zeros: $$2, 5, 1$$ Additionally, we are told that f(0) = -20.
Let's set up a polynomial function: $$f(x)=a(x - k_1)(x - k_2)(x - k_3)$$ Now, let's plug in for the zeros. The order does not matter: $$f(x)=a(x - 2)(x - 5)(x - 1)$$ $$f(x)=a(x^3 - 8x^2 + 17x - 10)$$ To find a, use the fact that f(0) is -20. $$-20=a(-10)$$ $$a=\frac{-20}{-10}=2$$ We know that a is 2: $$f(x)=2(x^3 - 8x^2 + 17x - 10)$$ $$f(x)=2x^3 - 16x^2 + 34x - 20$$

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