Chapter 0: Algebra II Review

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1 Chapter 0: Algebra II Review Topic 1: Simplifying Polynomials & Exponential Expressions p. 2 - Homework: Worksheet Topic 2: Radical Expressions p Homework: p. 45 #33-74 Even Topic 3: Factoring All Ways p Homework: p 68 & 69 #1-92 Even Topic 4: Rational Expressions p Homework: Worksheet Topic 5: Complex Fractions p Homework: Worksheet Topic 6: Completing the Square p Homework: p. 106 # (Complete the Square)

2 Name: Date: Period: Polynomial Operations: Addition/Subtraction: Combine like-terms only Chapter 0: Algebra II Review Topics 1: Simplifying Polynomials and Exponential Expressions Subtract from Multiplication: Every term by every term P a g e

3 Division: Exponent Rules: Remember: Exponents are always a little off from regular arithmetic rules. Addition/Subtraction: Combine coefficients of like-terms; exponents are unchanged Multiplication: Multiply coefficients; add exponents of like-bases Division: Divide coefficients; subtract exponents of like-bases Negative Exponents: I m stuck on the wrong side of the fraction line! Hint: deal with these first in complex questions! Fractional Exponents: Power over Root P a g e

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5 Homework: Perform the indicated operation. 1. (9y 2-12y + 5) - (12y 2 + 6y - 11) 2. 8(7r + y) - 3(5r - 2) 3. 2(y 2 + 4y) + 6y(y - 3) 4. (8r -1) - 3(10r - 8) 5. (3g 3-2g 2 + 1)(g - 4) 6. (9 - y 2 )(2y + 1) P a g e

6 Use your knowledge of exponent rules to simplify the following expressions P a g e

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8 Name: Date: Period: Do Now: Simplify the following radical expressions Chapter 0: Algebra II Review Topic 2: Radical Expressions Adding & Subtracting Radicals: Just like anything else, we can only When adding or subtracting radicals, both the AND the must be exactly the same Before we begin to combine, we must first. Example: Add: 1. Simplify each of the terms 2. Combine the like terms (add/subtract the coefficients of the like-radicands) P a g e

9 3. 4. Multiplying Radicals: Multiply the numbers outside the radicals the Multiply the numbers inside the radicals the Simplify the radicals in your final answers. Do not simplify until!!! Example: Multiply: 1. Multiply coefficients; Multiply Radicands 2. Simplify at the end *observe: if we simplified at the beginning, we d have to simplify again at the end! P a g e

10 Pairs of binomials like #10 are called: Definition: Conjugate Pairs - The result of multiplying conjugate pairs of radical expressions will ALWAYS be an INTEGER. When multiplying conjugate pairs, we can skip FOIL and just multiple first & last terms. Be VERY sure you are dealing with conjugate pairs before you take this shortcut! Example: 10 P a g e

11 Dividing Radicals: Divide the numbers outside the radicals, divide the numbers inside the radicals. Simplify the radicals in your final answers. If necessary, Example: Rationalizing Monomial Denominators Divide: 1. Divide as much as possible 2. Simplify at the end Rationalize if necessary Example: Rationalizing Binomial Denominators Divide: 1. Divide as much as possible (usually nothing is possible with binomial denominators) 2. Simplify at the end Rationalize if necessary P a g e

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13 Name: Date: Period: Factoring: Chapter 0: Algebra II Review Topic 3: Factoring ALL Ways When we factor it is important to remember that ; we are simply rewriting it in an equivalent form. GCF or Greatest Common Factor (review!): Factoring by GCF means that we what the terms have in. This can be a combination of numbers, variables, or both. Factor out by GCF P a g e

14 DOTS Factoring (review!): Another type of factoring stems from multiplication. This type of factoring is known as Difference of Two Squares or DOTS factoring. When we factor these types of expressions, we conjugate multiplication. Write each of the following binomials as the product of a conjugate pair Trinomial Case I Factoring (review!): Another type of factoring is trinomial factoring. This is when we have a trinomial. In Case I factoring, the leading coefficient is. To factor these, it is helpful to look at the term and the term. Write each of the following trinomials in factored form P a g e

15 Factor By Grouping (review!) We can factor by grouping when we have a polynomial that has. Class Example #1: Only one variable Steps: 1. Put terms in descending order, or with other like-factor terms 2. Group terms in sets of 2 3. Factor each group 4. Rearrange Use FOIL to check! Class Example #2: More than one variable Steps: 1. Put terms in descending order, or with other like-factor terms 2. Group terms in sets of 2 3. Factor each group 4. Rearrange Use FOIL to check 15 P a g e

16 Grouping Examples: Factor Trinomials - with a leading coefficient (review!) Case II Factoring To solve these examples, we use factoring by grouping by "splitting" up the middle into two factors. Always make sure that every example is written in standard form before you try to split & factor. Class Example #3 Steps: 1. Decide on factors & signs 2. Rewrite as 4 terms 3. Factor by grouping 4. Rearrange Use FOIL to check! Case II Factoring Examples: P a g e

17 Factoring Sums & Differences of Two Cubes General Rules be very aware of what signs are used & when! Worth studying and committing to memory. This will be used this year and even more in Calculus courses. Class Example #6: Steps: 1. Identify the cube roots of both terms 2. Plug in to the appropriate pattern (given above) Sum & Difference of Two Cubes Examples: Distribute carefully to check! P a g e

18 Name: Date: Period: Chapter 0: Algebra II Review Topic 4: Rational Expressions Define: Rational Expression: Domain: Limits to domain in the real number system Recall that the set of values which make up your domain is typically unlimited. You can plug any number you want in to an expression or a function. (The output depends on the rule given and can vary greatly and is only limited by the rule of the expression or function) There is one large notable, very important, exception to the domain of rational expressions. A denominator of a fraction can never!!!!!! Examples: Determine the numbers that must be excluded from each domain: 1) 2) 3) 18 P a g e

19 Simplifying Rational Expressions We must ensure we understand the structure of the expression we are simplifying. It is not proper to split up terms that are joined by addition or subtraction. Rather, we can only cancel factors that appear in both the numerator and denominator Examples: Simplify completely by factor & cancel : 1) 2) Review of Rational Expression Operations: Multiplication & Division To multiply rational expressions, factor all numerators and denominators completely. Then, across all numerator factors and all denominator factors, cancel any matches. Multiply what remains and ensure your final answer is in simplest form. To divide, FIRST, perform keep change reciprocal, then proceed the same way. Examples: Perform the indicated operation and simplify: 1) 2) 19 P a g e

20 3) 4) Addition & Subtraction Just like any type of adding or subtracting, we can only combine like terms. In a rational expression, like terms happen when we have COMMON DENOMINATORS. Therefore, we must get common denominators before we can combine. Review with arithmetic: Least common denominator: Find the LCD Multiply by what s missing to get common denominators Add across Practice the same method with a rational expression: Steps: 1) Change subtraction if necessary. 2) Find the LCD. 3) Multiply each term by what is missing. 4) Add across. 5) Simplify if possible. 20 P a g e

21 Examples: Perform the indicated operation and simplify: 1) 2) 3) 4) 21 P a g e

22 Homework: Simplify each rational expression: Indicate the values for which the rational expression is undefined: P a g e

23 Perform the indicated operation and simplify, if possible: P a g e

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25 Name: Date: Period: Simplifying Complex Fractions Chapter 0: Algebra II Review Topic 5: Complex Fractions Two methods for solving complex fractions are presented below. Each has a preferred time to be used, but both can be used in any situation with careful setup. Try both options to ensure you re fluent in both. Class Example #1: Single rational expressions in the numerator & denominator Rewrite left to right. Keep, change, reciprocal. Simplify. Class Example #2: Multiple rational expressions in the numerator and/or denominator STEPS: 1. Clean it up: Ensure ALL terms are fractions. 2. Find the least common denominator (LCD) of all of the fractions. We do this by factoring all of the denominators of the smaller fractions. 3. Multiply all of the fractions by the LCD we found in step 2. (By doing this, we are multiplying the numerator & denominator of our complex fraction by the LCD). ALL OF YOUR LITTLE DENOMINATORS MUST CANCEL! 4. Simplify whenever possible. 25 P a g e

26 Complex Fraction Examples: 1) 2) 3) 4) 26 P a g e

27 Complex Fractions Containing Radicals In order to simplify complex fractions with radicals, we must be able to multiply expressions with radicals There are 2 Cases: 1. When the expression under the radical is different from the expression without the radical: --> Here we cannot simplify further, so the expression is left as 2. When the expressions are the same inside and outside the radical: --> Here we will follow the following steps: 1. Rewrite the radical with exponents (no radical) 2. To multiply, we will add the exponents 3. Rewrite as a radical 27 P a g e

28 Examples: Complex Fractions with Radicals Simplify each of the following P a g e

29 Homework: Express each complex fraction or rational expression in simplest form: P a g e

30 Simplify each of the following P a g e

31 Name: Date: Period: Chapter 0: Algebra II Review Topic 6: Completing the Square Completing the Square: We will force the left-side of the equation to become a perfect square trinomial. Completing the Square 1. Move the constant term to the other side. 2. Be sure the coefficient of the highest power is one. If it is not, factor out the coefficient from 3. Create a perfect square trinomial by adding (to both sides!) Be careful if there was a constant factored out. 4. Factor the perfect square; add the constants together. 5. Isolate the variable to solve. (Square root both sides, remove what remains) Examples: P a g e

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