Chapter 5/6/7 Polynomials

6.1 Operations with Polynomials 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

Product of Powers Add exponents Examples:

Quotient of Powers Subtract exponents Examples:

Power of a Power Multiply exponents Examples

Power of a Product Raise each factor to the exponent Examples

Power of a Quotient Raise top and bottom by exponent Examples

Simplifying using several properties Examples

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

Multiplying Binomials FOIL First, Outer, Inner, Last Box Examples

Multiplying Polynomials Distributive property and combine like terms Examples

6.2 Dividing Polynomials

Polynomial by a Monomial Break into sum of separate fractions and simplify Examples

Polynomial by Binomial Long Division Synthetic Division Examples

5.3 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

7.4 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 No - One – No +

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.

7.5 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

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

7.6 Rational Exponents Two big rules

Examples X1/5*x7/5 X-3/4 X-2/3 6√16 3√2 6√4x4 Y1/2+1 Y1/2-1

7.7 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 3(5n-1)1/3-2=0 2+ √4x-4)≤6 √3x-6)+4≤7 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

5.4 Complex Numbers Big rules for Imaginary unit i

Powers of i √-18 √-32y3 -3i*2i √-12* √-2 Divide exponent by 4 Use remainder as new exponent Simplify i35 Simplify i55

Complex number a + bi a is real part, b is imaginary part

To find values of x and y that make equations true… Set real parts equal and solve Set imaginary parts equal and solve 2x-3+(y-4)i=3+2i

To add/subtract Add real parts Add imaginary parts (7-6i)+(9+11i)

To multiply complex numbers FOIL/Box and then simplify (6-4i)(6+4i) (4+3i)(2-5i) (7+2i)(9-6i)

Dividing complex numbers Multiply top and bottom by conjugate of bottom 2/(7-8i) (3-i)/(2-i) (2-4i)/(1+3i)

Solving equations using i 5x2+35=0 -5m2-65=0 -2m2-6=0 Find values of m and n to make equation true (3m+4)+(3-n)i=16-3i