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### Lesson 4-5 Fundamental Theorem of Algebra

Objective: To learn & apply the fundamental theorem of algebra & the linear factor theorem.

We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra.### The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

If f (x) is a polynomial of degree n, where n 1, then the equation f (x) = 0 has at least one complex root.

The Linear Factor Theorem

Just as an nth-degree polynomial equation has n roots, an nth-degree polynomial has n linear factors. This is formally stated as the Linear Factor Theorem.

The Linear Factor Theorem

If f (x) =anxn+ an-1xn-1+ … + a1x + a0 b, where n 1 and an 0 , then

f (x) = an (x - c1) (x - c2) … (x -cn)

where c1, c2,…, cn are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors.

This is the linear factorization for a fourth-degree polynomial.

f(x) = an(x - c1)(x - c2)(x - c3)(x - c4)

Use the given zeros: c1= -2, c2= 2, c3= i, and, from above, c4= -i.

= an(x + 2)(x -2)(x - i)(x + i)

= an(x2- 4)(x2+ 1) Multiply

f(x) = an(x4- 3x2- 4) Complete the multiplication

more

more

3.5: More on Zeros of Polynomial Functions

EXAMPLE: Finding a Polynomial Function with Given Zeros

Find a fourth-degree polynomial function f(x) with real coefficients that has

-2, 2, and i as zeros and such that f(3) = -150.

Solution Because iis a zero and the polynomial has real coefficients, the conjugate must also be a zero. We can now use the Linear Factorization Theorem.

EXAMPLE: Finding a Polynomial Function with Given Zeros

Find a fourth-degree polynomial function f(x) with real coefficients that has

-2, 2, and i as zeros and such that f(3) = -150.

Solution

f (3) = an(34 - 3 • 32- 4) = -150To find an, use the fact that f (3) = -150.

an(81 - 27 - 4) = -150Solve for an.

50an= -150

an= -3

Substituting -3 for an in the formula for f(x), we obtain

f(x) = -3(x4- 3x2- 4).

Equivalently,

f(x) = -3x4+ 9x2+ 12.

Multiplicity

- Multiplicity refers to the number of times that root shows up as a factor
- Ex: if -2 is a root with a multiplicity of 2 then it means that there are 2 factors :(x+2)(x+2)

Practice

- Find the polynomial that has the indicated zeros and no others:
- -3 of multiplicity 2, 1 of multiplicity 3
- Find the polynomial P(x) of lowest degree that has the indicated zeros and satisfies the given condition:
- 2 + 3i and 4 are roots, f(3) = -20
- Answer: f(x) = -16x2 + 58x - 104

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