Chapter 5

Polynomials

5.1 Monomials

Monomial- number, variable, or product of a number and one or more variables Cannot have: variables in denominator, variables with

negative exponents, or variables under radicals

Constants-numbers with no variables Coefficient – Numerical factor of monomial Degree- sum of the exponents of its variables Examples

Negative Exponents

nn

aa

1 n

na

a

1

Product of Powers

Add exponents Examples:

nmnm aaa

Quotient of Powers

Subtract exponents Examples:

nmn

m

aa

a

Power of a Power

Multiply exponents Examples

mnnm aa

Power of a Product

Raise each factor to the exponent Examples mmm baab

Power of a Quotient

Raise top and bottom by exponent

Examples

m

m

m

mm

m

mm

a

b

b

a

b

aand

b

a

b

a

,

Simplifying using several properties

Examples

5.2 Polynomials

Polynomial- Monomial or sum of monomials Each monomial is a term of the polynomial

Binomial- two terms Trinomial- three terms Degree of Polynomial- degree of term with

highest degree 3x2 + 7 3x5y3 – 9x4

Like terms – x and 2x

Simplifying Polynomials

Distribute across grouping symbols and combine like terms

Examples

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.

Multiplying Binomials

FOIL First, Outer, Inner, Last

Box Examples

Multiplying Polynomials

Distributive property and combine like terms Examples

5.3 Dividing Polynomials

Polynomial by a Monomial

Break into sum of separate fractions and simplify

Examples

Polynomial by Binomial

Synthetic Division Examples

5.4 Factoring Polynomials

Always look for GCF first!

Factoring binomials (2 terms)

Difference of two squares a² - b² = (a + b)(a – b)

Difference of two cubes a³ - b³ = (a – b)(a² + ab + b²)

Sum of two cubes a³ + b³ = (a + b)(a² - ab + b²)

Factoring trinomials (3 terms)

Perfect Square Trinomials a² +2ab+b² = (a + b)² a² -2ab+b² = (a - b)²

Factoring Trinomials

Grouping Four terms

Ax+bx+ay+by=x(a+b)+y(a+b)(x+y)(a+b)

Simplifying Quotients

Factor numerator and denominator completely

Simplify by cancelling

5.5 Roots of Real Numbers

Square root

Cube root

nth root

Index-Radical-Radicand

Principal Root

When there is more than one real root, the non-negative root is the principal root

Examples

Real Roots

n b>0 b<0 b=0

even One +

One -

No real roots

One real root, 0

odd One +

No -

One –

No +

One real root, 0

Simplifying Square Roots

Rules for Simplifying Radicals There are no perfect square, cubes, etc. factors

other than 1 under the radical. There isn’t a fraction under the radical. The denominator does not contain a radical

expression.

5.6 Radical Expressions

Product Property

Quotient Property

Rationalize the denominator means get the radical out of the bottom

Like radical expressions have the same index and same radicand

nnn baab

n

n

n

b

a

b

a

Radicals simplified if

Index small as possible No more factors of n in radicand No more fractions in radicand No radicals in denominator

5.7 Rational Exponents

Two big rules

mnn mn

m

bbb

nn bb 1

Examples

X1/5*x7/5

X-3/4

X-2/3

6√16 3√2

6√4x4

Y1/2+1

Y1/2-1

5.8 Radical equations and inequalities

Variables in radicand Undo square roots by squaring Undo cube roots by cubing Check for extraneous solutions

A number that does not satisfy the original equation

ALWAYS go back and check solutions Inequalities: must also solve for

Radicand ≥ 0

Examples

√x+1)+2=4 √x-15)=3- √x

Solution does not check No real solutions 3(5n-1)1/3-2=0 2+ √4x-4)≤6

Solve 4x-4≥0 first because even index so can’t be negative

Then solve complete inequality and test intervals √3x-6)+4≤7

On calculator

Set everything = 0 Graph Find “zero” (2nd calc 2:zero)

If inequality look for interval that is above or below the x-axis

5.9 Complex Numbers

Big rules for Imaginary unit i

12 i 1i

iiiiii 3210 ,1,,1

Powers of i

√-18 √-32y3

-3i*2i √-12* √-2 Divide exponent by 4 Use remainder as new exponent Simplify i35

Solve 3x2+48

Complex number

a + bi a is real part, b is imaginary part 2x-3+(y-4)i=3+2i

To find values of x and y that make equations true…

Set real parts equal and solve Set imaginary parts equal and solve

To add/subtract

Add real parts Add imaginary parts

To divide complex numbers

Multiply top and bottom by conjugate of bottom