Chapter 6 Study Guide

Sections: 6.2,6.3,6.4,6.5,6.6

Although section 6.8 is not on the study guide, it will be on the test.

Write each polynomial in factored form.

1. 2x3 +10x2 +12x

Factored Form of a Polynomial - 6.2

2. x4 − x3 − 6x2

3. − 3x3 +18x2 − 27x

Find the relative maximum, relative minimum and zeros of eachfunction.4. f x( ) = x3 − 7x2 +10x

Relative Maximum/Minimum and Zeros - 6.2

5. f x( ) = x3 − x2 − 9x + 9

End Behavior of a Function

If the degree of the polynomial is even, then the left behavior is the sameas the right behavior. If the degree of the polynomial is odd, then the left behavior is the opposite of the right behavior.

Left side of the graph

If the leading coefficient is positve, then the graph rises to the right. If theleading coefficient is negative, then the graph falls to the right.

Right side of the graph

ii) y = −x2 + 5x

iii) y = x3 iv) y = −x3 + 4x + 2

i) f x( ) = x4 − 3x3 + 5x

Examples

Factor Theorem

The expression x − a is a linear factor of a polynomial if and only if the value a is a zero of the polynomial function.

Theorem

i) f x( ) = x x + 4( ) x − 2( )Examples

ii) y = −x2 + 7x

Graph of a Polynomial Function - 6.2Find the zeros of each function. Then graph the function.7. f x( ) = −x x − 2( ) x + 5( )

-10

-5

10

5

105-5-10

Find the zeros of each function. State the multiplicity of multiple zeros.

8. y = x − 5( )3 x −1( )

Multiplicity of a Polynomial - 6.2

10. y = x4 − 8x3 +16x2

Divide using long division.

Polynomial Long Division - 6.3

2. 2x3 − 3x2 −18x − 8( )÷ x − 4( )

3. 9x3 − 48x2 +13x + 3( )÷ x − 5( )

Divide using synthetic division.

Synthetic Division - 6.3

5. −2x3 + 5x2 − x + 2( )÷ x + 2( )

6. x4 − 6x2 − 27( )÷ x + 2( )

Use synthetic division and the given factor to completely factor each polynomial.

7. f x( ) = x3 − 4x2 − 9x + 36; x + 3( )

Factoring Polynomials - 6.3

Remainder Theorem

If a polynomial P x( ) of degree n ≥1 is divided by x − a( ), where a is constant,then the remainder is P a( ).

Theorem

i) Find P 3( ) for P x( ) = x4 − 2x3 + x − 9.

Example

Use synthetic division and the remainder theorem to find P a( ).

Synthetic Division and Remainder Theorem - 6.3

10. P x( ) = 6x4 +19x3 − 2x2 − 44x − 24; a = − 23

Sum of Cubes

a3 + b3 = a + b( ) a2 − ab + b2( )

Sum and Difference of Cubes - 6.4

x3 +125 = x + 5( ) x2 − 5x + 25( )Example :

Difference of Cubes

a3 − b3 = a − b( ) a2 + ab + b2( )27x3 −1= 3x −1( ) 9x2 + 3x +1( )Example :

Factor the expression.

1. 8x3 − 343

Solving Polynomial Equations - 6.4Solve the equation.

2. 2x3 + 54 = 0

Solving Polynomial Equations - 6.4Solve each equation.

3. x4 − 5x2 + 4 = 0 4. x4 + 7x2 −18 = 0

5. 27 = −x4 −12x27. x3 + 3x2 − 4x −12 = 0

8. 4x4 − 6x2 − 4 = 6

9. The width of a box is 3 meters less than the length. The height is three times the length. The volume is 48 cubic meters. Find the length of the box.

Applications - 6.4

10. The sum of two positive numbers is 5 and the sum of their cubes is 35. What is the sum of their squares?

TheoremIf pq

is the simplest form and is a rational root of the polynomial

equation anxn + an−1x

n−1 + ...+ a1x1 + a0 = 0 with integer coefficients,

then p must be a factor of a0 and q must be a factor of an .

Rational Root Theorem - 6.5Use the Rational Root Theorem to list all possible rational roots for eachpolynomial equation. Then find any actual roots.

Finding Rational Roots - 6.5

2. x3 + 4x2 + x − 6 = 0

3. x4 + 2x2 −15 = 0

Find the roots of each polynomial equation.

Finding Rational Roots - 6.5

5. 5x3 − 24x2 + 41x − 20 = 0

6. 10x4 + x3 + 7x2 + x − 3= 0

Theorem: IrrationalLet a and b be rational numbers and b be an irrational number.

If a + b a is a root of a polynomial equation with rational

coefficients, then the conjugate a − b is also a root.

Irrational & Rational Root Theorem

Theorem: ImaginaryIf the imaginary number a + bi is a root of the polynomialequation with real coefficients, then the conjugate a − biis also a root.

Find a fourth-degree polynomial equation with integer coefficients thathas the given number of roots.

Finding Polynomial Equations - 6.5

8. 3 and 1− i

9. 2 and − 3i

TheoremIf P x( ) is a polynomial of degree n ≥1 with complex coefficients, then P x( ) = 0 has at least one complex root.

Fundamental Theorem of Algebra

i) x4 − 5x2 + 4 = 0 ii) x4 + 7x2 +12 = 0

zeros : zeros :

Type : Type :

CorollaryA polynomial of degree n cannot have more than n zeros.

For each equation, state the number of complex roots, the possible number of real roots, and the possible rational roots.

1. 2x5 − 4x4 − 4x2 + 5 = 0

Using theFundamental Theorem of Algebra - 6.6

3. − 2x6 − x2 + x − 7 = 0

Find all the zeros of each function.

4. f x( ) = x3 + 2x2 − 5x −10

Zeros of a Polynomial Function - 6.6

5. y = 4x3 + 9x2 + 22x + 5

6. f x( ) = x4 − 3x2 − 4