Warm up

Use the Rational Root Theorem to determine the Roots of :

x³ – 5x² + 8x – 6 = 0

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 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 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 Fundamental Theorem of Algebra

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.

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.

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.

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

= an(x + 2)(x -2)(x - i)(x + i) Use the given zeros: c1 = -2, c2 = 2, c3 = i, and, from above, c4 = -i.

f (x) = an(x - c1)(x - c2)(x - c3)(x - c4) This is the linear factorization for a fourth-degree polynomial.

= an(x2 - 4)(x2 + 1) Multiplyf (x) = an(x4 - 3x2 - 4) Complete the multiplication

moremore

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.

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

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

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

Solution

f (3) = an(34 - 3 • 32 - 4) = -150 To find an, use the fact that f (3) = -150. an(81 - 27 - 4) = -150 Solve for an.

50an = -150an = -3

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) = -20Answer: f(x) = -16x2 + 58x - 104