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**Intermediate Algebra 098A**

Review of Exponents & Factoring

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1.1 – Integer Exponents For any real number b and any natural number n, the nth power of b is found by multiplying b as a factor n times.

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**Exponential Expression – an expression that involves exponents**

Base – the number being multiplied Exponent – the number of factors of the base.

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Product Rule

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Quotient Rule

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Integer Exponent

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Zero as an exponent

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Calculator Key Exponent Key

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Sample problem

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more exponents Power to a Power

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Product to a Power

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Polynomials - Review Addition and Subtraction

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**Determine the coefficient and degree of a monomial**

Objective: Determine the coefficient and degree of a monomial

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Def: Monomial An expression that is a constant or a product of a constant and variables that are raised to whole –number powers. Ex: 4x xyz

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**Coefficient: The numerical factor in a monomial **

Definitions: Coefficient: The numerical factor in a monomial Degree of a Monomial: The sum of the exponents of all variables in the monomial.

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**Examples – identify the degree**

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**A monomial or an expression that can be written as a sum or monomials.**

Def: Polynomial: A monomial or an expression that can be written as a sum or monomials.

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**Def: Polynomial in one variable:**

A polynomial in which every variable term has the same variable.

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**Binomial: A polynomial containing two terms. **

Definitions: Binomial: A polynomial containing two terms. Trinomial: A polynomial containing three terms.

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**The greatest degree of any of the terms in the polynomial.**

Degree of a Polynomial The greatest degree of any of the terms in the polynomial.

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Examples:

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Objective Add and Subtract Polynomials

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**To add or subtract Polynomials**

Combine Like Terms May be done with columns or horizontally When subtracting- change the sign and add

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**Evaluate Polynomial Functions**

Use functional notation to give a polynomial a name such as p or q and use functional notation such as p(x) Can use Calculator

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**Calculator Methods 1. Plug In 2. Use [Table] 3. Use program EVALUATE**

4. Use [STO->] 5. Use [VARS] [Y=] 6. Use graph- [CAL][Value]

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**Apply evaluation of polynomials to real-life applications.**

Objective: Apply evaluation of polynomials to real-life applications.

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**Multiplication and Special Products**

Intermediate Algebra 5.4 Multiplication and Special Products

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Objective Multiply a polynomial by a monomial

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**Procedure: Multiply a polynomial by a monomial**

Use the distributive property to multiply each term in the polynomial by the monomial. Helpful to multiply the coefficients first, then the variables in alphabetical order.

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Law of Exponents

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**Multiply Special Products.**

Objectives: Multiply Polynomials Multiply Binomials. Multiply Special Products.

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**Procedure: Multiplying Polynomials**

1. Multiply every term in the first polynomial by every term in the second polynomial. 2. Combine like terms. 3. Can be done horizontally or vertically.

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**Multiplying Binomials**

FOIL First Outer Inner Last

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**Product of the sum and difference of the same two terms Also called multiplying conjugates**

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Squaring a Binomial

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**Use techniques as part of a larger simplification problem.**

Objective: Simplify Expressions Use techniques as part of a larger simplification problem.

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**Albert Einstein-Physicist**

“In the middle of difficulty lies opportunity.”

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**Intermediate Algebra –098A**

Common Factors and Grouping

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**A number or expression written as a product of factors.**

Def: Factored Form A number or expression written as a product of factors.

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**Greatest Common Factor (GCF)**

Of two numbers a and b is the largest integer that is a factor of both a and b.

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**Can do two numbers – input with commas and ). Example: gcd(36,48)=12**

Calculator and gcd [MATH][NUM]gcd( Can do two numbers – input with commas and ). Example: gcd(36,48)=12

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**Greatest Common Factor (GCF) of a set of terms**

Always do this FIRST!

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**Procedure: Determine greatest common factor GCF of 2 or more monomials**

1. Determine GCF of numerical coefficients. 2. Determine the smallest exponent of each exponential factor whose base is common to the monomials. Write base with that exponent. 3. Product of 1 and 2 is GCF

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**Factoring Common Factor**

1. Find the GCF of the terms 2. Factor each term with the GCF as one factor. 3. Apply distributive property to factor the polynomial

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**Example of Common Factor**

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**Factoring when first terms is negative**

Prefer the first term inside parentheses to be positive. Factor out the negative of the GCF.

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**Factoring when GCF is a polynomial**

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**Factoring by Grouping – 4 terms**

1. Check for a common factor 2. Group the terms so each group has a common factor. 3. Factor out the GCF in each group. 4. Factor out the common binomial factor – if none , rearrange polynomial 5. Check

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**Example – factor by grouping**

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**Ralph Waldo Emerson – U.S. essayist, poet, philosopher**

“We live in succession , in division, in parts, in particles.”

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**Intermediate Algebra 098A**

Special Factoring

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**a difference of squares a perfect square trinomial a sum of cubes **

Objectives:Factor a difference of squares a perfect square trinomial a sum of cubes a difference of cubes

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**Factor the Difference of two squares**

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Special Note The sum of two squares is prime and cannot be factored.

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**Factoring Perfect Square Trinomials**

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**Factor: Sum and Difference of cubes**

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Note The following is not factorable

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**Factoring sum of Cubes - informal**

(first + second) (first squared minus first times second plus second squared)

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**Intermediate Algebra 098A**

Factoring Trinomials of General Quadratic

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Objectives: Factor trinomials of the form

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Factoring 1. Find two numbers with a product equal to c and a sum equal to b. The factored trinomial will have the form(x + ___ ) (x + ___ ) Where the second terms are the numbers found in step 1. Factors could be combinations of positive or negative

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**Factoring Trial and Error**

1. Look for a common factor 2. Determine a pair of coefficients of first terms whose product is a 3. Determine a pair of last terms whose product is c 4. Verify that the sum of factors yields b 5. Check with FOIL Redo

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Factoring ac method 1. Determine common factor if any 2. Find two factors of ac whose sum is b 3. Write a 4-term polynomial in which by is written as the sum of two like terms whose coefficients are two factors determined. 4. Factor by grouping.

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Example of ac method

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Example of ac method

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**Factoring - overview 1. Common Factor 2. 4 terms – factor by grouping**

3. 3 terms – possible perfect square 4. 2 terms –difference of squares Sum of cubes Difference of cubes Check each term to see if completely factored

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Isiah Thomas: “I’ve always believed no matter how many shots I miss, I’m going to make the next one.”

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**Intermediate Algebra 098A**

Solving Equations by Factoring

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**If a and b are real numbers**

Zero-Factor Theorem If a and b are real numbers and ab =0 Then a = 0 or b = 0

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**Example of zero factor property**

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**Solving a polynomial equation by factoring.**

Factor the polynomial completely. Set each factor equal to 0 Solve each of resulting equations Check solutions in original equation. Write the equation in standard form.

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**Example – solve by factoring**

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**Example: solve by factoring**

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**Example: solve by factoring**

A right triangle has a hypotenuse 9 ft longer than the base and another side 1 foot longer than the base. How long are the sides? Hint: Draw a picture Use the Pythagorean theorem

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Solution Answer: 20 ft, 21 ft, and 29 ft

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**Example – solve by factoring**

Answer: {-1/2,4}

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**Example: solve by factoring**

Answer: {-5/2,2}

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**Example: solve by factoring**

Answer: {0,4/3}

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**Example: solve by factoring**

Answer: {-3,-2,2}

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Sugar Ray Robinson “I’ve always believed that you can think positive just as well as you can think negative.”

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Maya Angelou - poet “Since time is the one immaterial object which we cannot influence – neither speed up nor slow down, add to nor diminish – it is an imponderably valuable gift.”

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