High School Algebra II Unlocked (2016)

GOALS

By the end of this chapter, you will be able to

•Determine how many roots a polynomial has, based on the degree of the polynomial

•Factor polynomials with imaginary number coefficients

•Create and solve systems of equations to solve single-variable polynomial equations

•Identify and apply direct, inverse, joint, and combined variations

•Perform arithmetic operations with rational expressions

•Rewrite rational expressions as the sum of a polynomial and a remainder expression

Lesson 2.1. The Fundamental Theorem of Algebra

REVIEW

A complex number is a number in the form a + bi, where a and b are both real numbers and i is the imaginary number defined as being equal to .

The roots of a single-variable polynomial expression are the solutions for the variable when the polynomial is set equal to 0.

QUADRATIC FORMULA: x = for a quadratic equation of the form ax² + bx + c = 0

In Chapter 1, we saw that single-variable polynomials with real-number coefficients can have real roots, imaginary roots, or both.

The Fundamental Theorem of Algebra asserts that any single-variable polynomial of degree n with complex coefficients has exactly n roots, including all instances of repeated roots.

Every single-variable polynomial of degree 1 or greater with complex coefficients has at least one complex root.

The single-variable polynomial 1/4 x⁴ − 1/4 x³ − 3x² − x + 4 is of degree 4 (the highest power of x in the polynomial), and it has exactly 4 roots (−2, −2, 1, and 4), even though −2 is a repeated root. This is a case where all coefficients are real and all roots are real. Real numbers are a subset of complex numbers, so the Fundamental Theorem of Algebra applies in these cases, but it also applies when coefficients and/or roots include imaginary numbers. 2x⁴ − 30x² − 32, another polynomial of degree 4, has exactly 4 roots (−4, 4, −i, and i), even though two of them are imaginary.

A single-variable
polynomial of degree 1
or greater is a polynomial
that includes one variable,
such as x, whether to
the first power (such
as in 7x − 1) or to a
higher power (such as
in −2x² or x⁹ + 6x⁴).

How many roots does 3z² + 4z + 5 have? What are the roots?

This is a polynomial of degree 2, so it has two roots, according to the Fundamental Theorem of Algebra.

The polynomial does not seem factorable, so we’ll use the quadratic formula,

The two roots can also be written in standard form as −2/3 + i and −2/3 − i.

The standard form of a complex number is a + bi, where a and b are real.

You can use the quadratic formula to find the roots of any polynomial of degree 2, which will always produce two roots (because of the ± sign), even if they are the same root (in cases of root multiplicity) or complex roots (as in Example 1). This echoes the Fundamental Theorem of Algebra.

Here is how you may see the Fundamental Theorem of Algebra on the SAT.

Which of the following represents a polynomial function with roots of {−2, −2, and 5}?

A) f(x) = x³ + x² − 16x + 20

B) f(x) = x³ − x² − 16x − 20

C) f(x) = x² + 3x − 10

D) f(x) = x² − 3x − 10

So far, we have only looked at real polynomials. A real polynomial is a polynomial with real coefficients for all of its terms.

An irreducible quadratic is a real quadratic that has no real roots, but even so, it still has two complex roots, as in Example 1. An irreducible quadratic cannot be factored unless you use imaginary numbers in the factors.

Every real polynomial can be factored into a product of real linear factors and real irreducible quadratic factors.

Billy noticed that the polynomials he factored for homework always factored into real linear factors, with at most one real irreducible quadratic. He proposed a theory that every real polynomial has at most two roots that are not real. Graciela disproved his theory with the counterexample of 2x⁴ + 7x² + 3. Show how her example disproves Billy’s theory.

The polynomial 2x⁴ + 7x² + 3 is a real polynomial, because the coefficients 2, 7, and 3 are real numbers. To factor this polynomial, recognize that the trinomial is in the form 2a² + 7a + 3, which factors as (2a + 1)(a + 3).

2(x²)² + 7(x²) + 3 = (2x² + 1)(x² + 3)

Both factors are irreducible quadratics. The roots of 2x² + 1 are and − . The roots of x² + 3 are i and −i . So, all four roots of 2x⁴ + 7x² + 3 are imaginary numbers. Graciela’s example is a real polynomial that has more than two roots that are not real.

The Fundamental Theorem of Algebra covers polynomials with “complex coefficients,” so it applies even when one or more of the coefficients is not real.

How many roots does the polynomial x³ − 2ix² + 15x have? What are the roots?

This is a polynomial of degree 3, so it has three roots.

The x term has an i in its coefficient, so we should rewrite the constant term, 15, in terms of i.

The three roots of x³ − 2ix² + 15x are 0, −3i, and 5i.

Alternatively, after factoring out x to get x(x² − 2ix + 15), we could use the quadratic formula to solve for the roots of x² − 2ix + 15.

If the coefficient of x²
or the constant in a
quadratic contains an
imaginary number, the
quadratic formula may
result in a very ugly pair of
expressions, including
. We will not deal with
those situations here, but
know that any resulting
solutions (roots) can still
be written in the form
a + bi, the standard form
of a complex number,
where a and b are real.

The Fundamental Theorem of Algebra tells us that a single-variable polynomial with three roots is of degree three, or a cubic, even in the case that one root is repeated. You can eliminate (C) and (D).

Write the factor that corresponds to each given root: the factor that will equal 0 if x is equal to the given root. Then multiply all three factors.

The correct answer is (B).