PreCalculus 300. Algebra 2 Review

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1 PreCalculus 00 Algebra Review Algebra Review The following topics are a review of some of what you learned last year in Algebra. I will spend some time reviewing them in class. You are responsible for knowing these topics and coming to see me for any etra help you may need. Factoring. GCF (Greatest Common Factor) The GCF is always the first thing you look for when factoring. There may be additional factoring you will need to do after the GCF. Even if there isn t a GCF, you may still need to factor the polynomial. Do not assume that if there isn t a GCF the polynomial is prime. Make sure you check your answer by distributing. 6 y z y z 9 9 y 6y z 8z. Difference of Squares (DOS) In order to have DOS, you must have eactly terms, both of which are perfect squares. All eponents of variables must be even. (Even though 9 is a perfect 9 square, isn t.) You also have to have subtraction. There is no such factoring as sum of squares. The way to factor DOS is a b ( a b)( a b). y z 9z c 5 5 d ( 7) 9. Trinomials One way to factor trinomials ( a b c) is by using the sum-product rule. When a, you find numbers that multiply to the c value and add to the b value. Don t forget to look for a GCF first! 5 yz 9yz 0 yz If you do have an a term other than, you can either guess-and-check or split the middle term Multiply a and c.. Find numbers that multiply to the number you got in step # and add to b.. Split the middle term and rewrite the trinomial according to the numbers you came up with in step #. If one of the numbers in step # is positive and the other is negative, it s better to write the negative term before the positive one.

2 PreCalculus 00 Algebra Review. Group the first terms and the last terms with parentheses. 5. Factor each set of parentheses using GCF. Whatever remains after taking the GCF should be the same. 6. Regroup by taking each GCF in set of parentheses and whatever remains in the other set. 7. Check your work by FOILing! Sum/Difference of Cubes (S/DOC) In order to have S/DOC, you must have eactly terms, both of which are perfect cubes. All eponents of variables must be multiples of. (Even though 8 8 is a perfect cube, isn t.) Unlike DOS, you may have addition or subtraction. The way to factor S/DOC is a b ( a b)( a ab b ) or a b ( a b)( a ab b ). If you factor correctly, the quadratic (trinomial) part of S/DOC factoring should not be factorable. Look for a GCF first! Here are some additional practice problems. 5 z 6z y 0 y z 5z y 6 5 z 08 z 0a 7 ab 5b 9 Completing the Square (CTS) CTS is used to take a quadratic equation ( y a b c) and make it look like y k a( h). Take a look at y 6.. Make sure a.. Send the c term to the opposite side of the equation.. Take the b term, divide it by, and then square the quotient. This number will be added to both sides of the equation in order to make a perfect square trinomial (PST) on one side.. Factor the side of the equation with the PST. 5. Simplify the other side of the equation to make it look like y k.

3 PreCalculus 00 Algebra Review y 0 7 y y 0 Take a look at y 6 6. Here, a.. Send the c term to the opposite side of the equation.. Since a, we need to make it by taking out the GCF.. Take the b term, divide it by, and then square the quotient. This number will be added to both sides of the equation in order to make a perfect square trinomial (PST) on one side. When adding this number to the other side of the equation, multiply it by the GCF.. Factor the side of the equation with the PST. If you do it correctly, it should look like ( h). 5. Simplify the other side of the equation to make it look like y k. y 6 y y 7 0 y y 0 y 9

4 PreCalculus 00 Algebra Review Rational Eponents and Eponent Laws Let s first recall the properties m n mn Product Property: a a a 5 Eample: a Quotient Property: a 5 Eample: m mn a where a 0 n 5 m mn Power of a Power Property: a a Eample: n Power of a Quotient Property: Eample: y y a b m a b m m where b 0 m m Power of a Product Property: ab a b Eample: Eample: m y z y z y z 8 n n Negative Eponents: b and b n n b b Eample: and a 0 Zero Eponents: Eample: 0 Definition of Rational Eponents: Eample: m n b n m b m n b

5 PreCalculus 00 Algebra Review Now you practice!! If you need etra review try visiting Rewrite in radical form and simplify y Rewrite in eponential form and simplify. Use the smallest base possible... y a y Simplify completely y

6 PreCalculus 00 Algebra Review y 7y y y 8. 6a b a b 9b y 6. y y y w y z

7 PreCalculus 00 Algebra Review Radicals Simplifying Radicals: We always want the smallest possible number under the radical, otherwise known as the radicand. Look for factors that are Perfect Squares, Perfect Cubes, Perfect Fourths, etc. The types of factors you look for need to match the inde of the radical y z 7 y z Multiplying Radicals Eample: Multiply everything under the radical together and then simplify! Eample: Multiply everything outside of the radical together and everything under the radical together and then simply!. y y. ( 8 y 6 y ) ( 8 y ) Adding & Subtracting Radicals: You can only add and subtract radicals when they are like radicals. How do you define like radicals? z z z 8z 8 8 7

8 PreCalculus 00 Algebra Review Dividing Radicals: Radicals are not allowed in the Denominator: Rationalize it! Multiplying a fraction by a convenient form of. ** When rationalizing the denominator, multiply the numerator and denominator by what will make it a perfect root! Fractions with comple denominators Conjugates: We can create a conjugate of any binomial. 6 7 With a binomial in the denominator, multiply the fraction by a form of that uses the conjugate. ***** When you multiply by the conjugate, the radical goes away! Additional resource: 8

9 PreCalculus 00 Algebra Review Now it is your turn! Simplify the following; epress all answers in lowest terms with no negative eponents y 7 y 5 y 8y y 9 y y

10 PreCalculus 00 Algebra Review Radical Equations Simple Radical Equation: e Comple Radical Equation: ve There are five steps to solve a radical equation.. Isolate the radical. Make sure the radical equals a positive number. Raise each side to a power to get rid of the radical. Solve 5. Check!. 5 c. p For equations with fractional eponents:. Isolate the fractional eponent. Raise to a power that will cancel the fractional eponent. Simplify and repeat step if necessary. Solve 5. Check!. 6 m m 5. k 0 6. w 7 w 0

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12 PreCalculus 00 Algebra Review Factor Theorem: If Rational Root Theorem a is a factor of f, then 0 f a. Rational Root Theorem: If a rational number p q is the root of a polynomial equation, n n n a b c... y z 0, then p is a factor of the constant (z) and q is a factor of the leading coefficient (a). The purpose of the Rational Root Theorem is to determine the possible roots of a higher degree function. Eample: f 7 a) Find all possible roots using the Rational Root Theorem. b) Use your calculator to find the actual roots (roots that make the equation equal 0) c) Now use synthetic division and other factoring methods to break down the polynomial into smaller factors and find ALL eact roots. (You may need to use the Quadratic Formula at the end.) For further eplanation, watch this video: y_mode=&safe=active

13 PreCalculus 00 Algebra Review Practice: Find all eact roots (real and imaginary).. y 5 6. y 5 5. y y 8 5. y y y y

14 PreCalculus 00 Algebra Review Solving Rational Equations Recall that a rational epression is a fraction where the numerator and/or denominator is a polynomial. To simplify a rational epression: Factor the numerator completely Factor the denominator completely Cancel any common factors Eample: 5a b 5ab 5ab = Recall that when multiplying two fractions, we simplify and then multiply across, numerator numerator, denominator denominator. Same idea when multiplying rational epressions. Eample: a a a a a When dividing two fractions, we multiply by the reciprocal of the second fraction. Same idea when dividing rational epressions. Eample: y 6y y y 8 Recall that when adding fractions, we find a common denominator first, rewrite the fractions with the common denominator, then add the numerators. Same idea when adding rational epressions.. Factor the denominators. Determine the Least Common Denominator (LCD). Multiply in the missing pieces of the LCD. Simplify the numerators 5. Add numerators 6. Check to see if numerator is factorable, then simplify Eample: 5 5

15 PreCalculus 00 Algebra Review When subtracting rational epressions, the only difference is in Step #5 above, where you subtract numerators instead of adding them! Eample: We use the techniques above when solving rational equations. Two types:. One rational epression is equal to another rational epression. Method of solving: CROSS MULTIPLY. Sum or Difference of rational epressions is equal to another rational epression. Method of solving: MULTIPLY EVERYTHING BY THE LCD! (LCM) ***Always remember to check your solutions For further clarification: Solve and check for etraneous solutions

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( 3) ( 4 ) 1. Exponents and Radicals ( ) ( xy) 1. MATH 102 College Algebra. still holds when m = n, we are led to the result

( 3) ( 4 ) 1. Exponents and Radicals ( ) ( xy) 1. MATH 102 College Algebra. still holds when m = n, we are led to the result Exponents and Radicals ZERO & NEGATIVE EXPONENTS If we assume that the relation still holds when m = n, we are led to the result m m a m n 0 a = a = a. Consequently, = 1, a 0 n n a a a 0 = 1, a 0. Then