Summer Math Packet Precal

Summer Math Packet In preparation for Pre-Calculus, we have prepared a packet of concepts that students should know how to do. These concepts have been taught in previous math classes. This packet does not require you to use a calculator; in fact you should not use a calculator at all on any of these problems. Pre-Calculus builds on the concepts in this packet. We start teaching Pre-Calculus concepts on the first day of school. We expect you to know the concepts in the packet in order to help you be successful in Pre-Calculus. If you are struggling with this packet, get help from a friend, parent, or tutor. If you can’t find someone to help you, there are tutors available. A list of tutors can be found by calling Reagan High School. Keep in mind these tutors may charge a fee. You will have a graded assignment within the first two weeks of school. The graded assignment will cover all the concepts in the packet, but will not be the exact same problems. The graded assignment will be without a calculator. 1. ( x + 7 ) Multiplying Polynomials No Calculator!!! 2 2) ( x − 11) 2 3. ( x + 4 ) 3 4. ( x + h ) 3 ( 5. ( x + 1) x 2 − 3 x − 4 ( ) 6. ( x + h ) x 2 + 3 xh + 8 7. (a + b ) 2 ) Factoring No Calculator!! Factor each polynomial completely. If the polynomial cannot be factored write prime. 1) 2 x 2 − 128 2) x 2 − 10 x + 24 3) a 3 − 64b 3 4) 5 x 2 + 40 x − 10 5) 2 x 2 − 11x + 12 6) x 3 + 16 x 2 + 64 x 7) x 3 + 3 x 2 − 4 x − 12 8) 24 x 2 − 54 9) 6 x 3 − 18 x 2 10) 5c 2 + 4cd − d 2 11) 27 y 3 + 125 12) 20 x 2 − 4 x − 72 13) − x 2 + 100 14) 4 x 4 − 64 15) a 4 − 2a 2 + 1 16) 9 x 3 + 12 x 2 − 45 x 17) n 2 − 2 n − np + 2 p 18) 24 x 2 + 4 x − 60 Adding and Subtracting Fractions No Calculator!!! Simplify each expression. 1. 2 5 + 3 7 2. 1 5 − 6 18 3. 6 +5 x 4. 3x −7 4y 5. 3 4 − x2 x 6. x 7x + 2 x + 5 x − 25 7. 6 4 1 + − 5 x 9 x 3x 8. 8 2 + x − 4x + 4 x − 2 9. x 5 + x − 9 4 x − 12 10. 5x x+5 + x−5 x+2 2 11. 3 4 − x + 3 3x 2 Multiplying and Dividing Fractions No Calculator!!! Simplify each expression. 4 2 1. ⋅ 5 3 11 4. 7 7 − 18 7. 4 x ⋅ 13 7 8 10. 3 x 5x 7 2 3. 7 4 9 1 3 2. ⋅ − 9 7 5. 8. 2 3 5 x 6. 5 3 − x+2 −7 ⋅ 5x 4 x 9. 11. 7x + 2 5x − 3 9x + 4 6x + 7 − 11 ⋅ 9x 10 x 12. 2 5 3 7 14. 1 4− 7 y 13. z 7 Remember you cannot cancel at the beginning!!! 1 1+ x 15. 1 1− x 2+ x −4 3 16. x +7 3 1) 2) 3) 4) 5) 6) 7) 2 3− 2 7 3+4 4+ 3 2− 3 2+ 2 6+ 2 3i − 2 5i − 3 6−i 2 6+i 2 3 + 7i 7i Rationalize the denominator No Calculator!! Solve Quadratic Equations No Calculator!! Find all real and imaginary solutions for all problems. Solve the following by factoring. x 2 = 3x + 4 2) 9 x = 10 x 2 1) 4) x ( x − 5 ) = 36 5) 5(4 x − 3) = 30 2 10) ( y + 4 )2 2 Solve the following by completing the square. 11) x 2 + 10 = 8 x 12) 8x2 + 2x = 1 8) (2 x − 5)2 13) 2 x 2 + 16 x + 39 = 0 ( x − 6 )( x − 8) = 24 Solve the following by using the square root property. 3x 2 + 2 = 0 7) (x + 5)2 − 12 = 0 6) 9) 3) x2 − 5x + = −11 = 18 41 =0 4 Solve the following using the Quadratic Formula. You should have the Quadratic Formula memorized. 14) 3x 2 = 2 − 9 x 15) 5 x 2 − 2 x = −4 16) 12 x 2 = x + 6 Find the domain of functions No Calculator!! State the domain of each function using interval notation. 1) f (x ) = 2 x − 5 2) f (x ) = x 5− x 4) f (x ) = 3x 2 − 4 x + 9 5) f (x ) = x x+4 7) f (x ) = 1 3x − 27 2 1 8) f ( x ) = x 2 − 10 x + 24 3) 6) f (x) = 4 x + 5 f (x ) = − 2 x + 5 Remember the quadratic formula!!! Solve each rational equation. Rational Equations No Calculator!!! 1. x 2 = x−3 5 2. 4 = 3. 2 3x − 1 + =4 x x+3 4. 4x − 3 x+6 = 6− x−2 x+2 5. 2 6 3 + 2 = x + 5 x − 25 x − 5 6. 13 x + 20 4 6 − = x + 13 x + 42 x + 6 x + 7 2 5 2 + x 3 Logarithms No Calculator!!! Write each equation in logarithmic form. 2. 5 −3 = 1. 4 2 = 16 1 125 Write each equation in exponential form. 3. log 3 81 = 4 4. log 49 7 = 1 2 Evaluate each expression. 5. log 100 6. log 2 32 8. log 64 4 9. log 5 58 7. log 3 1 81 Solve each equation. 10. log 7 x = 3 13. log 3 1 =x 27 11. log 8 (5 x − 11) = 2 12. log x 6 = ( ) 14. log 4 x + 3 = log 4 5 x 2 1 2 15. log 125 = 3 log x 16. 2 log 9 3 − log 9 5 = log 9 x 17. log 4 x + log 4 2 = 3 18. log 3 ( x + 1) − log 3 ( x − 1) = 4 ANSWER KEYS 1. ( x + 7 ) Multiplying Polynomials No Calculator!!! 2 x 2 + 14 x + 49 2) ( x − 11) 2 x 2 − 22 x + 121 3. ( x + 4 ) 3 x 3 + 12 x 2 + 48 x + 64 4. ( x + h ) 3 x 3 + 3 x 2 h + 3 xh 2 + h 3 ( 5. ( x + 1) x 2 − 3 x − 4 ) x3 − 2x 2 − 7x − 4 ( 6. ( x + h ) x 2 + 3 xh + 8 ) x 3 + 4 x 2 h + 3 xh 2 + 8 x + 8h 7. (a + b ) 2 a 2 + 2ab + b 2 Factoring--ANSWERS No Calculator!! Factor each polynomial completely. If the polynomial cannot be factored write prime. 1) 2 x 2 − 128 2) x 2 − 10 x + 24 3) a 3 − 64b 3 2 ( x + 8)( x − 8) ( x − 4 )( x − 6 ) ( a − 4b ) ( a 2 + 4ab + 16b 2 ) 4) 5 x 2 + 40 x − 10 5 ( x2 + 8x − 2) 5) 2 x 2 − 11x + 12 ( 2 x − 3)( x − 4 ) 6) x 3 + 16 x 2 + 64 x x(x+8)(x+8) 7) x 3 + 3 x 2 − 4 x − 12 ( x + 3)( x + 2 )( x − 2 ) 8) 24 x 2 − 54 6 ( 2 x + 3)( 2 x − 3) 9) 6 x 3 − 18 x 2 6 x 2 ( x − 3) 10) 5c 2 + 4cd − d 2 11) 27 y 3 + 125 12) 20 x 2 − 4 x − 72 13) − x 2 + 100 14) 4 x 4 − 64 15) a 4 − 2a2 + 1 ( 5c − d )( c + d ) ( 3 y + 5) ( 9 y 2 − 15 y + 25) 4 ( 5 x + 9 )( x − 2 ) (10 − x )(10 + x ) 4 ( x 2 + 4 ) ( x + 2 )( x − 2 ) ( a + 1)( a − 1)( a + 1)( a − 1) 16) 9 x 3 + 12 x 2 − 45 x 3x ( 3x − 5)( x + 3) 17) n 2 − 2 n − np + 2 p ( n − 2 )( n − p ) 18) 24 x 2 + 4 x − 60 4 ( 3x + 5)( 2 x − 3) Adding and Subtracting Fractions No Calculator!!! Simplify each expression. 1. 2 5 + 3 7 2. 29 21 4. 3x −7 4y − 5. 3 x − 28 y 4y 7. 6 4 1 + − 5 x 9 x 3x 59 45 x 10. 5x x+5 + x−5 x+2 6 x 2 + 10 x − 25 x 2 − 3 x − 10 1 5 − 6 18 3. 6 + 5x x 1 9 3 4 − x2 x 6. 3 − 4x x2 8. 6 +5 x x 7x + 2 x + 5 x − 25 x 2 + 2x x 2 − 25 8 2 + x − 4x + 4 x − 2 2 9. x 5 + x − 9 4 x − 12 2 9 x + 15 4 x 2 − 36 2 x−2 11. 3 4 − x + 3 3x 5 x − 12 3x 2 + 9 x Multiplying and Dividing Fractions No Calculator!!! Simplify each expression. 1. 4 2 ⋅ 5 3 2. 8 15 11 4. 7 7 − 18 − 7. 198 49 4 x ⋅ 13 7 4x 91 1 3 ⋅− 9 7 − 5. 1 21 2 3 5 x 6. 5 3 2 15 9. − 7 x − 14 20 x 2 11 ⋅ 9x 10 99x 10 11. 7x + 2 5x − 3 9x + 4 6x + 7 − − 42 x 2 − 61x − 14 45 x 2 − 7 x − 12 56 15 x 2 x 2 5 x 15 x+2 −7 ⋅ 5x 4 x 8 10. 3 x 5x 7 12. 9 14 − − 8. 2 3. 7 4 9 y 13. z 7 3 7 14. 1 4− 7 2+ 5x 2 y 7z Remember you can not cancel at the beginning!!! 1 1+ x 15. 1 1− x x +1 x −1 17 27 x −4 3 16. x +7 3 x − 12 x + 21 Rationalize the denominator--ANSWERS No Calculator!! 1) 2) 3) 4) 5) 6) 7) 2 3− 2 7 3+4 4+ 3 2− 3 2+ 2 6+2 2 7 − 21 + 4 7 13 11 + 6 3 6+ 2 5+ 2 2 17 3i − 2 5i − 3 21 + i 34 6−i 2 6+i 2 17 6 2 − i could write the answer all over 19 as one fraction 19 19 3 + 7i 7i 7 − 3i 7 Solve Quadratic Equations--ANSWERS No Calculator!! Find all real and imaginary solutions for all problems. Solve the following by factoring. x 2 = 3x + 4 2) 9 x = 10 x 2 1) 9 X= − 1,4 x= 0, 10 x ( x − 5 ) = 36 X= 9,−4 4) 5) 8) (2 x − 5)2 x=2, 12 i 6 X= ± 3 X= 8x2 + 2x = 1 1 1 x= − 2 , 4 ( x − 6 )( x − 8) = 24 Solve the following by using the square root property. 6) 3x 2 + 2 = 0 7) (x + 5)2 − 12 = 0 9) 3) x= − 5 ± 2 3 5(4 x − 3) = 30 2 10) 3± 6 4 ( y + 4 )2 2 = −11 x= 5 ± i 11 2 = 18 x=10, -2 Solve the following by completing the square. 11) x 2 + 10 = 8 x X= 4 ± 6 41 =0 4 5 5 ± 4i x= 2 ± 2i or 2 12) x2 − 5x + 13) x= − 4 ± 2 x 2 + 16 x + 39 = 0 i 14 2 or − 8 ± i 14 2 Solve the following using the Quadratic Formula. You should have the Quadratic Formula memorized. 14) 3x 2 = 2 − 9 x 15) 5 x 2 − 2 x = −4 16) 12 x 2 = x + 6 −9 ± 105 1 + i 19 3 2 x= x= x = ,− 6 5 4 3 Find the domain of functions--ANSWERS No Calculator!! State the domain of each function using interval notation. 1) f (x ) = 2 x − 5 5 ,∞ 2 4) f (x ) = x 5− x f (x ) = 3x 2 − 4 x + 9 5) f (x ) = (− ∞, ∞) x x+4 6) (− ∞,−4) ∪ (− 4, ∞) f (x ) = 1 3x − 27 2 (− ∞,−3) ∪ (− 3,3) ∪ (3, ∞) f (x) = 4 x + 5 3) (− ∞,5) ∪ (5, ∞) (− ∞, ∞) 7) 2) f (x ) = − 2 x + 5 − ∞, 1 8) f ( x ) = x 2 − 10 x + 24 (− ∞,4) ∪ (4,6) ∪ (6, ∞) 5 2 Remember the quadratic formula!!! Solve each rational equation. 1. x 2 = x−3 5 Rational Equations No Calculator!!! 2. 4 = 3 2 -2 3. 2 3x − 1 + =4 x x+3 4. − 11 ± 145 2 5. 2 6 3 + 2 = x + 5 x − 25 x − 5 -19 5 2 + x 3 4x − 3 x+6 = 6− x−2 x+2 9 ± 105 2 6. 13 x + 20 4 6 − = x + 13 x + 42 x + 6 x + 7 2 44 3 Logarithms No Calculator!!! Write each equation in logarithmic form. 2. 5 −3 = 1. 4 2 = 16 log 4 16 = 2 log 5 1 125 1 = −3 125 Write each equation in exponential form. 3. log 3 81 = 4 4. log 49 7 = 1 2 1 3 4 = 81 49 2 = 7 Evaluate each expression. 5. log 100 2 6. log 2 32 7. log 3 5 8. log 64 4 1 3 1 81 -4 9. log 5 58 8 Solve each equation. 10. log 7 x = 3 343 13. log 3 11. log 8 (5 x − 11) = 2 12. log x 6 = 15 1 =x 27 36 ( ) 14. log 4 x + 3 = log 4 5 x 2 1 2 -3 15. log 125 = 3 log x 5 17. log 4 x + log 4 2 = 3 32 64 5 16. 2 log 9 3 − log 9 5 = log 9 x 9 5 18. log 3 ( x + 1) − log 3 ( x − 1) = 4 41 40