Lesson 2.5, page 312Zeros of Polynomial Functions

Objective: To find a polynomial with specified zeros, rational

zeros, and other zeros, and to use Descartes’ rule.

Introduction

Polynomial Type of Coefficient

5x3 + 3x2 + (2 + 4i) + i complex

5x3 + 3x2 + √2x – π real

5x3 + 3x2 + ½ x – ⅜ rational

5x3 + 3x2 + 8x – 11 integer *******

Rational Zero Theorem If the polynomial

f(x) = anxn + an-1xn-1 + . . . + a1x + a0

has integer coefficients, then every rational

zero of f(x) is of the form p

qwhere p is a factor of the constant a0

and q is a factor of the leading coefficient an.

Rational Root (Zero) Theorem

• If “q” is the leading coefficient and “p” is the constant term of a polynomial, then the only possible rational roots are + factors of “p” divided by + factors of “q”. (p / q)

• Example: • To find the POSSIBLE rational roots of f(x), we

need the FACTORS of the leading coefficient (6 for this example) and the factors of the constant term (4, for this example). Possible rational roots are

41246)( 35 xxxxf

factors of 1, 2, 4 1 1 1 2 41,2,4, , , , ,

factors of 1, 2, 3, 6 2 3 6 3 3

p

q

See Example 1, page 313.

• Check Point 1: List all possible rational zeros of f(x) = x3 + 2x2 – 5x – 6.

Another example

• Check Point 2: List all possible rational zeros of f(x) = 4x5 + 12x4 – x – 3.

How do we know which possibilities are really zeros (solutions)?

• Use trial and error and synthetic division to see if one of the possible zeros is actually a zero.

• Remember: When dividing by x – c, if the remainder is 0 when using synthetic division, then c is a zero of the polynomial.

• If c is a zero, then solve the polynomial resulting from the synthetic division to find the other zeros.

See Example 3, page 315.

• Check Point 3: Find all zeros of

f(x) = x3 + 8x2 + 11x – 20.

Finding the Rational Zeros of a Polynomial1. List all possible rational zeros of the

polynomial using the Rational Zero Theorem.

2. Use synthetic division on each possible rational zero and the polynomial until one gives a remainder of zero. This means you have found a zero, as well as a factor.

3. Write the polynomial as the product of this factor and the quotient.

4. Repeat procedure on the quotient until the quotient is quadratic.

5. Once the quotient is quadratic, factor or use the quadratic formula to find the remaining real and imaginary zeros.

Check Point 4, page 316

• Find all zeros of

f(x) = x3 + x2 - 5x – 2.

How many zeros does a polynomial with rational coefficients have?

• An nth degree polynomial has a total of n zeros. Some may be rational, irrational or complex.

• Because all coefficients are RATIONAL, irrational roots exist in pairs (both the irrational # and its conjugate). Complex roots also exist in pairs (both the complex # and its conjugate).

• If a + bi is a root, a – bi is a root• If is a root, is a root.• NOTE: Sometimes it is helpful to graph the

function and find the x-intercepts (zeros) to narrow down all the possible zeros.

ba ba

See Example 5, page 317.

• Check Point 5

• Solve: x4 – 6x3 + 22x2 - 30x + 13 = 0.

Fundamental Theorem of Algebra (page 318)

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

• Note: This theorem just guarantees a zero exists, but does not tell us how to find it.

Linear Factorization Theorem, pg. 319

Remember…• Complex zeros come in pairs as

complex conjugates: a + bi, a – bi

• Irrational zeros come in pairs.

a c b , a c b

Practice

Find a polynomial function

(in factored form) of degree 3

with 2 and i as zeros.

More Practice

Find a polynomial function

(in factored form) of degree 5

with -1/2 as a zero with multiplicity 2,

0 as a zero of multiplicity 1,

and 1 as a zero of multiplicity 2.

Example

• Find a 4th-degree polynomial function f(x) with real coefficients that has -2, 2 and i as zeros and such that f(3) = -150.

Extra Example

Suppose that a polynomial function of degree 4 with rational coefficients has i and -3 +√3 as zeros. Find the other zero(s).