CHAPTER 3 Toolbox. Integer Exponents  If a is a real number and n is a positive integer, then a n represents a as a factor n times in a product  a n.

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Presentation transcript:

CHAPTER 3 Toolbox

Integer Exponents  If a is a real number and n is a positive integer, then a n represents a as a factor n times in a product  a n = aaa…aa  a is called the base  n is called the exponent

Properties of Exponents  For any real number a, and integers m and n,  a m x a n = a m+n  a m ÷ a n = a m-n

Examples

Zero and Negative Exponents

Examples

Absolute Value  Absolute value measures the distance between the number inside the absolute value, and zero  Always a positive answer  If number inside is positive, absolute value does nothing  If number inside is negative, absolute vale makes it positive

Examples

Rational Exponents

Examples

Multiplication on Monomials and Binomials  Monomials – Polynomials with only one term  Binomials – Polynomials with two terms  If multiplying monomials together, multiply like terms together  If multiplying a monomial with a binomial, multiply the monomial by each term in the binomial  If multiplying binomials together, use FOIL method

Examples

Special Binomial Products  (x+a)(x-a) = x 2 – a 2 Difference of Two Squares  (x+a) 2 = x 2 + 2ax + a 2 Perfect Square Trinomial  (x-a) 2 = x 2 – 2ax + a 2 Perfect Square Trinomial

Examples

Factoring  When factoring always factor out the Greatest Common Factor (GCF), if one exists

Factoring  After GCF, use knowledge of special binomial products to factor

Factoring  If there are 4 terms, factor by grouping

Factoring  If a trinomial is being factored, follow the following steps to factor StepsExample To factor a quadratic trinomial in the variable x:Factor 5x – 6 + 6x 2 1.Arrange the trinomial with the powers of x in descending order 6x 2 +5x Form the product of the second-degree term and the constant term (first and third terms) 6x 2 (-6) = -36x 2 3.Determine if there are 2 factors of the product in step above that will sum to the middle term of quadratic (if there are no such factors, trinomial cannot be factored) -36x 2 = (-4x)(9x) and -4x + 9x = 5x 4.Replace the middle term from step 1 with the sum of the two factors from step 3 6x 2 +5x – 6 = 6x 2 -4x + 9x Factor the four term polynomial by grouping6x 2 -4x + 9x – 6 = (6x 2 -4x) + (9x – 6) = 2x(3x-2) + 3(3x-2) = (3x-2)(2x+3)

Example

Complex Numbers

Examples

Identify Complex Numbers  A + Bi  Real = A  Imaginary = A + Bi  Pure Imaginary = Bi

Homework  Page 172  1-31