Lesson Objectives
  • Learn about the Fundamental Theorem of Algebra
  • Learn about the Complete Factorization Theorem
  • Learn about the Number of Zeros Theorem
  • Learn how to write a polynomial function given zeros and a point

How to Write a Polynomial Function Given Real Zeros and a Point


In this lesson, we will learn about the Fundamental Theorem of Algebra, the Complete Factorization Theorem, and the Number of Zeros Theorem. We will then learn how to write a polynomial function when we are given the real zeros and a point.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra tells us that if f(x) is a polynomial function of degree n with complex coefficients: $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$ Where: $$n ≥ 1, a_n ≠ 0$$ Then the equation f(x) = 0 has at least one complex zero. Since all real numbers are also complex numbers, this theorem applies to any polynomial function with real coefficients.

Complete Factorization Theorem

We can use the Fundamental Theorem of Algebra with the factor theorem to show that a polynomial function can be factored completely into linear factors. If f(x) is a polynomial function of degree n, where n is 1 or more, then there exist complex numbers: $$a, k_1, k_2, ... , k_n$$ Where: $$a ≠ 0$$ Such that: $$f(x) = a(x - k_1)(x - k_2) \cdots (x - k_n)$$

Proof

From the Fundamental Theorem of Algebra, if f(x) is of degree n, where n is 1 or more, then there is some number k1 such that: $$f(k_1) = 0$$ Using the Factor Theorem: $$f(x) = (x - k_1)q_1(x)$$ If q1(x) is of degree 1 or more, then we can apply the same thought process. There will be some number k2 such that: $$q_1(k_2) = 0$$ Using the Factor Theorem: $$f(x) = (x - k_1)(x - k_2)q_2(x)$$ Since f(x) is of degree n, we repeat this process n times to obtain our model below: $$f(x) = a(x - k_1)(x - k_2) \cdots (x - k_n)$$ Each of these factors gives us a zero for f(x), so f(x) of degree n, where n is 1 or more, has n zeros, given by: $$k_1, k_2, ..., k_n$$ Theses zeros need not all be different. The number of times a factor appears in the factored form of the equation is known as the multiplicity. This result leads us to the Number of Zeros theorem.

Number of Zeros Theorem

A polynomial function of degree n ≥ 1 has at most n distinct zeros. Additionally, this polynomial function of degree n ≥ 1 will have exactly n zeros, provided that a zero of multiplicity k is counted k times. For example, suppose we have the following function. $$f(x) = (x + 2)^2$$ This function is of degree 2, so how many zeros are we going to have? $$f(x) = (x + 2)^2$$ $$f(-2) = 0$$ It would appear at first glance that this function only has one zero but notice that it is a zero that occurs twice. $$f(x) = (x + 2)^2 = (x + 2)(x + 2)$$ $$= (x - (-2))(x - (-2))$$ This is what we mean by a zero of multiplicity 2.
As another example, consider the function g(x). $$g(x) = (x + 1)(x - 3)^2$$ $$= (x - (-1))(x - 3)(x - 3)$$ Here, we have the following zeros: $$-1, 3 (\text{multiplicity }2)$$ Again, we can see that the zero of 3 occurs twice. We can use the theorems presented to write a polynomial function given the real zeros and a point. Let's look at some examples.
Example #1: Find a function f defined by a polynomial of degree 3 that satisfies the given conditions. $$\text{Zeros of: }2, 5, 1$$ $$f(0) = -20$$ 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. $$-20 = a(0^3 - 8(0)^2 + 17(0) - 10)$$ $$-20=a(-10)$$ $$a=\frac{-20}{-10}=2$$ We know that a is 2: $$f(x)=2(x^3 - 8x^2 + 17x - 10)$$ $$f(x)=2x^3 - 16x^2 + 34x - 20$$ Example #2: Find a function f defined by a polynomial of degree 3 that satisfies the given conditions. $$\text{Zeros of: }{-1}, 2, 4$$ $$f(0) = 12$$ 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: $$f(x)=a(x - (-1))(x - 2)(x - 4)$$ $$f(x)=a(x + 1)(x - 2)(x - 4)$$ $$f(x)=a(x^3 - 5x^2 + 2x + 8)$$ To find a, use the fact that f(0) is 12. $$12 = a(0^3 - 5(0)^2 + 2(0) + 8)$$ $$12=a(8)$$ $$a=\frac{12}{8}=\frac{3}{2}$$ We know that a is 3/2: $$f(x)=\frac{3}{2}(x^3 - 5x^2 + 2x + 8)$$ $$f(x)=\frac{3}{2}x^3 - \frac{15}{2}x^2 + 3x + 12$$ Example #3: Find a function f defined by a polynomial of degree 4 that satisfies the given conditions. $$\text{Zeros of: }{-1}, 2, 8$$ Where -1 is a zero of multiplicity 2. $$f(1) = 140$$ Let's set up a polynomial function: $$f(x)=a(x - k_1)(x - k_2)(x - k_3)(x - k_4)$$ Now, let's plug in for the zeros. The order does not matter: $$f(x)=a(x - (-1))(x - (-1))(x - 2)(x - 8)$$ $$f(x)=a(x + 1)(x + 1)(x - 2)(x - 8)$$ $$f(x)=a(x + 1)^2(x - 2)(x - 8)$$ $$f(x)=a(x^4 - 8x^3 - 3x^2 + 22x + 16)$$ To find a, use the fact that f(1) is 140. $$140 = a(1^4 - 8(1)^3 - 3(1)^2 + 22(1) + 16)$$ $$140 = a(1 - 8 - 3 + 22 + 16)$$ $$140 = a(28)$$ $$a=\frac{140}{28}=5$$ We know that a is 5: $$f(x)=5(x^4 - 8x^3 - 3x^2 + 22x + 16)$$ $$f(x)=5x^4 - 40x^3 - 15x^2 + 110x + 80$$

Skills Check:

Example #1

Write a polynomial function of degree 3 that satisfies the given conditions. $$\text{Zeros}: 4, 2, -5$$ $$f(0)=40$$

Please choose the best answer.

A
$$f(x)=-7x^3 + 5x^2 - 8x + 1$$
B
$$f(x)=x^3 - 3x^2 + 9x + 1$$
C
$$f(x)=x^3 - x^2 - 22x + 40$$
D
$$f(x)=-2x^3 - 3x^2 - 11x + 6$$
E
$$f(x)=2x^3 - 3x^2 - 11x + 6$$

Example #2

Write a polynomial function of degree 3 that satisfies the given conditions.

Please choose the best answer. $$\text{Zeros}: 3 \hspace{.2em}\text{mult.}\hspace{.2em}2, 1$$ $$f(0)=-9$$

A
$$f(x)=-5x^3 + 2x^2 + 3x - 4$$
B
$$f(x)=-x^3 + 5x^2 + 2x + 1$$
C
$$f(x)=x^3 + 7x^2 + 5$$
D
$$f(x)=x^3 - 7x^2 + 15x - 9$$
E
$$f(x)=2x^3 - 3x^2 + 5x + 1$$

Example #3

Write a polynomial function of degree 3 that satisfies the given conditions. $$\text{Zeros}: -1, 2, 4$$ $$f(0)=8$$

Please choose the best answer.

A
$$f(x)=3x^3 - 8x^2 + 16x + 80$$
B
$$f(x)=\frac{1}{2}x^3 - \frac{5}{2}x^2 + x + 4$$
C
$$f(x)=2x^3 - 10x^2 + 20x + 16$$
D
$$f(x)=x^3 - 5x^2 + 2x + 8$$
E
$$f(x)=2x^3 + 10x^2 - 20x - 40$$
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