8 th Grade Honors Variable Manipulation Part 3 Student

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1 8 th Grade Honors Variable Manipulation Part 3 Student 1 MULTIPLYING BINOMIALS-FOIL To multiply binomials, use FOIL: First, Outer, Inner, Last: Example: (x + 3)(x + 4) First multiply the First terms: x x x = x 2 Next the Outer terms: x x 4 = 4x. Then the Inner terms: 3 x x = 3x. And finally the Last terms: 3 x 4 = 12. Then add and combine like terms: x 2 + 4x + 3x + 12 = x 2 + 7x If (x + r) 2 = x x + r 2 for all real numbers x, then r =? 2. The expression (3x 4y 2 )(3x + 4y 2 ) is equivalent to: 3. The expression (4z + 3)(z 2) is equivalent to: FACTORING OUT A COMMON DIVISOR A factor common to all terms of a polynomial can be factored out. All three terms in the polynomial 3x x 2-6x contain a factor of 3x. Pulling out the common factor yields 3x(x 2 + 4x - 2). Remember that if you factor a term out completely, you are still left with 1: in the expression 6x 2 + 9x + 3, you can factor a 3 out of everything. You're left with 3(2x 2 + 3x + 1). 4. x 2 x + px 5. 36x 2-64y x x 2 + 9x

2 FACTORING THE DIFFERENCE OF SQUARES One of the test maker's favorite classic quadratics is the difference of squares. a 2 - b 2 = (a - b)(a + b) Example: x 2-9 factors to (x - 3)(x + 3). 2 FACTORING THE SQUARE OF A BINOMIAL There are two other classic quadratics that occur regularly on the ACT: a 2 + 2ab + b 2 = (a + b) 2 a 2-2ab + b 2 = (a - b) 2 For example, 4x x + 9 factors to (2x + 3) 2 and n 2-10n, + 25 factors to (n - 5) 2. Recognizing a classic quadratic can save a lot of time on Test Day-be on the lookout for these patterns. (HINT: Any time you have a quadratic and one of the numbers is a perfect square, you should check for one of these three patterns.) 7. For all n, (3n+ 5) 2 =? 8. For all x,(3x + 7) 2 =? 9. ( 1 3 a b)2 =? FACTORING OTHER POLYNOMIALS-FOIL IN REVERSE To factor a quadratic expression, think about what binomials you could use FOIL on to get that quadratic expression. To factor x 2-5x + 6, think about what First terms will produce x what Last terms will produce + 6, 2 and what Outer and Inner terms will produce -5x. If there is no number in front of the first term, you are looking for two numbers that add up to the middle term and multiply to the third term. So here, you'd want two numbers that add up to -5 and multiply to 6. (Pay attention to sign-negative vs. positive makes a big difference here!) The correct factors are (x - 2)(x - 3). You can also solve for x for each factor. In this case x = 2 and x = 3. Note: Not all quadratic expressions can be factored. These expressions will be covered in Math 2.

3 10. What is the product of the two real solutions of the equation 2x = 3 - x 2? Which of the following is a factored form of the expression 5x 2 13x 6? a. (x 3)(5x + 2) b. (x 2)(5x 3) c. (x 2)(5x + 3) d. (x + 2)(5x 3) e. (x + 3)(5x 2) 12. If x 2 45b 2 = 4xb, what are the 2 solutions for x in terms of b? 13. What values of x are solutions for x 2-2x = 8? 14. What is the product of the 2 solutions of the equation x 2 + 4x 21 = 0? 15. Which of the following is a polynomial factor of x 2 2x 24? a. x 4 b. x + 4 c. x + 6 d. 6 - x e. x 16. What is the smallest value of x that satisfies the equation x(x + 4) = -3?

4 4 SIMPLIFYING AN ALGEBRAIC FRACTION Simplifying an algebraic fraction is a lot like simplifying a numerical fraction. The general idea is to find factors common to the numerator and denominator and cancel them. Thus, simplifying an algebraic fraction begins with factoring, which often involves reverse-foil. To simplify x2 x 12, first factor the numerator and denominator: x 2 x 12 x 2 9 x 2 9 = (x 4)(x+3) (x 3)(x+3) Canceling x + 3 from the numerator and denominator leaves you with (x 4) (x 3). 17. If x is any number other than 3 and 6, then (x 3)(x 6) (3 x)(x 6) =? 18. For x 2 169, (x 13)2 x =? 19. For all a 0 and b 0, a+b b(a+b) 2a(a+b) =? 20. If A + B = 7A+2B and A, B, and x are integers greater than 1, than what must x equal? x 21. For all x 1, x2 2x+1 x 1 is equal to?

5 5 ADDING ALGEBRAIC FRACTIONS (GREATEST COMMON FACTOR IN DENOMINATOR) To add fractions, the denominators must be the same. Therefore, as a common denominator choose the LCM of the original denominators. The least common denominator must be a multiple of the denominator of each of the fractions. First find the least common denominator. Then, convert each fraction to an equivalent fraction with the same denominator by multiplying each fraction by the missing factors. Finally, add the fractions together and reduce as necessary. Example: Add 3 ab + 4 bc + 5 cd Solution: The least common denominator is abcd because each of these variables are included in at least one of the denominators. To change 3 into an equivalent fraction with denominator abcd, simply multiply ab by ab the factors it is missing, namely cd. Therefore, we must also multiply 3 by cd. That accounts for the first term in the numerator. To change 4 into an equivalent fraction with denominator abcd, multiply bc by the factors bc it is missing, namely ad. Therefore, we must also multiply 4 by ad. That accounts for the second term in the numerator. To change 5 into an equivalent fraction with denominator abcd, multiply cd by the factors it is cd missing, namely ab. Therefore, we must also multiply 5 by ab. That accounts for the last term in the numerator. Remember that EACT factor of the original denominators must be a factor of the common denominator ab bc cd = 3cd+4ad+5ab abcd Example: When adding fractions, a useful first step is to find the least common denominator (LCD) of the fractions. What is the LCD for these fractions? 2 a. 3 x 5 x 7 x 11 b. 3 2 x 5 2 x 7 x 11 c. 3 2 x 5 2 x 11 3 d. 3 2 x 5 2 x 7 x 11 3 e. 3 2 x 5 3 x 7 x 11 4, 13, x5 5 2 x7x11 3x11 3 Solution: The least common denominator must be a multiple of the denominator of each of the three given fractions. Take the factors of each denominator (3, 5, 7, 11) to the highest powers they appear: 3 2 (first fraction), 5 2 (second fraction), 7 (second fraction), and 11 3 (third fraction). This results in 3 2 x 5 2 x 7 x 11 3, choice D. Choice E is the product of all three denominators and its wrong because choice D is smaller and still a multiple of all three denominators; in other words, both are common denominators but D is the least. 22. What is the least common denominator for the expression below? a 2 x b x c b 2 x c b x c 2

6 23. What is the solution for the expression below? 4y 6(x 2)(x+5) + 2y 3x(x+5) What is the solution for the expression below? 3-2 4(x 2 1) (x 1)(x 2) SOLVING A SYSTEM OF EQUATIONS You can solve for two variables only if you have two distinct equations. If you have one variable, you only need one equation, but if you have two variables, you need two distinct equations. Two forms of the same equation will not be adequate. If you have three variables, you need three distinct equations, and so on. Combine the equations in such a way that one of the variables cancels out. To solve the two equations 4x + 3y = 8 and x + y = 3, multiply both sides of the second equation by -3 to get: - 3x - 3y = -9. Now add the equations; the 3y and the -3y cancel out, leaving x = -1: 4x + 3y = 8 +(-3x - 3y = -9) X = -1 Plug that back into either one of the original equations and you'll find that y = If mn = k and k =x 2 n, and nk 0, which of the following is equal to m? a. 1 b. 1/x c. x d. x e. x What is the largest possible product for 2 odd integers whose sum is equal to 32?

7 27. When x/y = 4, x 2 12y 2? The larger of two numbers exceeds twice the smaller number by 9. The sum of twice the larger and 5 times the smaller number is 74. If a is the smaller number, which equation below determines the correct value of a? a. 5(2a + 9) + 2a = 74 b. 5(2a - 9) + 2a = 74 c. (4a + 9) + 5a = 74 d. 2(2a + 9) + 5a = 74 e. 2(2a - 9) + 5a = The average of a and b is 6 and the average of a, b, and c is 11. What is the value of c? 30. The sum of the real numbers a and b is 13. Their difference is 5. What is the value of ab? 31. If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y =? 32. If x + y = 13 and 2y = 16, what is the value of x?

8 8 33. The costs of carriage rides of different lengths, given in half miles, are shown in the table below: Number of half miles Cost $8.00 $8.50 $9.00 $10.50 Each cost consists of a fixed charge and a charge per half mile. What is the fixed charge? 34. Marcia makes and sells handcrafted picture frames in 2 sizes: small and large. It takes her 2 hours to make a small frame and 3 hours to make a large frame. The shaded triangular region shown below is the graph of a system of inequalities representing weekly constraints Marcia has in making the frames. For making and selling s small frames and l large frames. For making and selling s small frames and l large frames, Marcia makes a profit of 30s + 70l dollars. Marcia sells all the frames she makes. For every hour that Marcia spends making frames in the second week of December each year, she donates $3 from that week s profit to a local charity. This year, Marcia made 4 large frames and 2 small frames in that week. What is the percent of that week s profit (rounded to the nearest percent) Marcia donated to the charity?

9 Answer Key x 2 16y z 2 5z x(x + 1 p) 5. 4(9x 2-16y 4 ) 6. 3x(x 2 + 9x + 3) 7. 9n n x x a2 2 ab + b (x 3)(5x + 2) 12. 9b or -5b and x (x 13) (x+13) b 2a x a 2 x b 2 x c yx 4y 6x(x 2)(x+5) (x 2) 8(x+1) 4(x+1)(x 1)(x 2) 25. x y (2a + 9) + 5a = $ %