UNIT 5.6 THE FUNDAMENTAL UNIT 5.6 THE FUNDAMENTAL THEOREM OF ALGEBRATHEOREM OF ALGEBRA

Warm UpIdentify all the real roots of each equation.

0, –2, 41. 4x5 – 8x4 – 32x3 = 0

–1, 1, –4, 4

2. x3 –x2 + 9 = 9x

0, 5

3. x4 + 16 = 17x2

4. 3x3 + 75x = 30x2

1, –3, 3

Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots.

Identify all of the roots of a polynomial equation.

Objectives

You have learned several important properties about real roots of polynomial equations.

You can use this information to write polynomial function when given in zeros.

Write the simplest polynomial function with the given zeros.

If r is a zero of P(x), then x – r is a factor of P(x).

P(x) = (x + 2)(x – 2)(x – 4)

P(x) = (x2 – 4)(x – 4)

P(x) = x3– 4x2– 4x + 16

Multiply the first two binomials.

Multiply the trinomial by the binomial.

Check It Out! Example 1a

–2, 2, 4

Write the simplest polynomial function with the given zeros.

If r is a zero of P(x), then x – r is a factor of P(x).P(x) = (x – 0)(x – )(x – 3) 2

3

P(x) = x3– x2+ 2x11 3

Multiply the first two binomials.

Multiply the trinomial by the binomial.

Check It Out! Example 1b

0, , 32 3

P(x) = (x2 – x)(x – 3) 2 3

Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

Using this theorem, you can write any polynomial function in factor form.

To find all roots of a polynomial equation, you can use a combination of the Rational Root Theorem, the Irrational Root Theorem, and methods for finding complex roots, such as the quadratic formula.

p = –20, and q = 1.

Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots.

The polynomial is of degree 4, so there are exactly four roots for the equation.

Step 1 Use the rational Root Theorem to identify rational roots.

±1, ±2, ±4, ±5, ±10, ±20

Check It Out! Example 2

Find the real roots at or near –5 and 1.

Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots.

Check It Out! Example 2 Continued

Test –5. The remainder is 0, so (x + 5) is a factor.

Step 3 Test the possible real roots.

Check It Out! Example 2 Continued

1 4 –1 16 –20–5

–5

1 –1 4 –4 0

5 –20 20

Test 1 in the cubic polynomial. The remainder is 0, so (x – 1) is a factor.

The polynomial factors into (x + 5)(x3 – x2 + 4x – 4) = 0.

Check It Out! Example 2 Continued

1 –1 4 –4 1

1

1 0 4 0

0 4

Step 4 Solve x2 + 4 = 0 to find the remaining roots.

The polynomial factors into (x + 5)(x – 1)(x2 + 4) = 0.

x2 + 4 = 0

x2 = –2x = ±2i

The fully factored form of the equation is (x + 5) (x – 1)(x + 2i)(x – 2i) = 0. The solutions are –5, 1, –2i, +2i).

Check It Out! Example 2 Continued

Write the simplest function with zeros 2i, , and 3.

Step 1 Identify all roots.

Check It Out! Example 3

By the Rational Root Theorem and the Complex Conjugate Root Theorem, the irrational roots and complex come in conjugate pairs. There are five roots: 2i, –2i, , , and 3. The polynomial must have degree 5.

1 21 2

1+ 2

Step 2 Write the equation in factored form.

Step 3 Multiply.

P(x) = x5 – 5x4+ 9x3 – 17x2 + 20x + 12

Check It Out! Example 3 Continued

P(x) = [ x - (2i)][x + (2i)][x - ( 1 + x )][x - (1 - x )](x - 3)

A grain silo is in the shape of a cylinder with a hemisphere top. The cylinder is 20 feet tall. The volume of the silo is 2106 cubic feet. Find the radius of the silo.

List the important information:

• The cylinder is 20 feet tall.

• The height of the hemisphere is x.

• The volume of the silo is 2106 cubic feet.

11 Understand the ProblemThe cylinder and the hemisphere will have the same radius x. The answer will be the value of x.

Check It Out! Example 4

22 Make a Plan

Write an equation to represent the volume of the body of the silo.

V = Vhemisphere + Vcylinder

V(x) = x3 + 20x2 2 3

Set the volume equal to 2106.

x3 + 20x2 = 2106 2 3

Vhemisphere = ( r3) and Vcylinder = x2h .

4 3

1 2

Solve33

x3 + 20x2 – 2106 = 02 3

Write in standard form.

Divide both sides by . x3 + 20x2 – 2106 = 02 3

6

26 234 0

234 21062 3

9 20 0 –21062 3

The graph indicates a positive root of 9. Use synthetic division to verify that 9 is a root, and write the equation as (x – 9)( x2 + 26x + 234) = 0. The radius must be a positive number, so the radius of the silo is 9 feet.

2 3

Look Back44

Substitute 6 feet into the original equation for the volume of the silo.

V(9) = (9)3 + 20(9)22 3

V(9)= 2106

All rights belong to their respective owners.Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism,

comment, news reporting, TEACHING, scholarship, and research.

Fair use is a use permitted by copyright statute that might otherwise be infringing.

Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.