Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) = 3x. f x = 4x 5 x+ 8 3. f ( x) = 7x + x+ 4 x 1 f x = x + x + x 4. 1 9 6 4 1 5. f ( x) = x+ 5 6. f ( x) = 5 6x 3x + 1. Let f x x x = 3 4 + 7 (a) Find each of the following without using your calculator. f () f ( 1) (b) Find each of the following using the Table feature of your calculator. f (7.34) f ( 5.73) 3 3. Let f ( x) = x 5x 6x+ 9 and gx = 6x + x 3. Find each of the following and simplify your answer by combining like terms. (a) f ( x) + g( x) (b) f ( x) g( x) (c) f( x) 3 gx Page 1 of 16
3. Multiply the polynomials. Simplify your answer and write each result in descending order. (a) (x 3)( x+ 5) (d) (4x 5)(4x+ 5) (b) 3 ( x 4)( x 6) + + (e) ( x + 4) (c) ( x )(3x 4x 7) + (f) (x 3) 4. The average base pay Bt ()(in thousands of dollars) of an MBA graduate is modeled by the function (a) Find B (0). Bt () =.3t 1.1t+ 98, where t is the number of years since 000. (b) Write a sentence explaining the meaning of B (0). (c) Find B (7). (d) Write a sentence explaining the meaning of B (7). Page of 16
Function Graph x- intercepts Factors Solve f ( x ) = 0 f x = x 1 f x = 4x 8x 3 f ( x ) = x x 6 4 f ( x ) = x 4 Page 3 of 16
Function Graph x- intercepts Factors Solve f ( x ) = 0 f x = x 6x+ 9 5 6 f x = 6x 19x+ 10 7 3 f x = x 4x 6x 8 f ( x ) = x + 1 page 33 Putting It All Together Page 4 of 16
Factoring Polynomials (5.3 5.5) Factoring Special Products a a b = ( a + b)( a b) Difference of Two Squares + b Sum of Two Squares does not factor over the reals + ab + b = ( a b ) ab + b = ( a b ) Perfect Square Trinomials 3 + 3 = + + Sum of Two Cubes a + a a b a b a ab b a b a b a ab b = + + Difference of Two Cubes 3 3 General Guidelines for Factoring: 1. Factor GCF (if it exists).. For a binomial (two terms), try using the difference of two squares formula. 3. For a second-degree trinomial ax + bx+ c : a. If a = 1, then try to find two integers whose product is c and whose sum is b. b. If a 1, then try to factor using the AC-Method. 4. For an expression with four terms, try factoring by grouping. 5. Continue applying steps 4 until the polynomial is completely factored. Zero Factor Property: Let A, B, and C be real numbers. If AB = 0, then A= 0 or B = 0. In other words, if the product of two [or more] numbers is zero, then at least one of the numbers is zero. Solving Equations by Factoring: If an equation can be solved by factoring, we solve it by the following steps: 1. Write the equation so that one side of the equation is zero.. Factor the nonzero side of the equation. 3. Apply the zero factor property. 4. Solve the equation that results from applying the zero factor property. Page 5 of 16
5.6 Polynomial Equations 1. Solve the equations graphically. a. f ( x ) = 0 b. f ( x ) = 6. Solve the equations symbolically. a. 4x + 5 = 0x b. 4 4x = x + 18 3. page 36 #68 4. page 36 #74 5. Solve the equation using the ZERO and INTERSECT features of your graphing calculator. 3 x + = x Page 6 of 16
8.1 & 8.: Quadratic Functions and Their Graphs A quadratic function is a function whose equation can be put into the form f ( x) ax ( h) k a. = + (vertex form) or f ( x) = ax + bx+ c (standard form) where 0 Which of the following are examples of quadratic functions? 1) f ( x) = x + 3x 5 4) f( x) = ( x 1) + 10 x ) f ( x) = 5x 5 5) f ( x) = 3() + 4 3 3) f ( x) = x 3x 4 6) f x = 5 x + 4 A parabola is the graph of a quadratic function. f x x x = 4 + f x x x = + 4 6 Page 7 of 16
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1 a) b) c) d) e) f) g) i) increasing: decreasing: a) b) c) d) e) f) g) i) increasing: decreasing: 3 a) b) c) d) e) f) g) i) increasing: decreasing: Page 9 of 16
4 a) b) c) d) e) f) g) i) increasing: decreasing: 5 a) b) c) d) e) f) g) i) increasing: decreasing: 6 a) b) c) d) e) f) g) i) increasing: decreasing: Page 10 of 16
x x x x + 6x+ 9= + 10x+ 5 = 8x+ 16 = 14x+ 49 = x x x x x x x x x x x x + 1 + = + 6 + 16 + = + 8 10 + = 5 18 + = 9 Completing the Square: Vertex Form: Page 11 of 16
Simplifying Radical Expressions (7.) Product Property: Quotient Property: Simplifying Radical Expressions: A radical expression k, k 0 is considered simplified when there are no perfect square factors other than 1, and there are no radicals in the denominator. Simplify: 1. 4. 81 3. 4 5 4. 5 8.3: Quadratic Equations Symbolic techniques for solving quadratic equations include factoring squarerootproperty QuadraticFormula completingthesquare Page 1 of 16
Solve the quadratic equations using the method indicated. Factoring Square Root Method Quadratic Formula Completing the Square x + 4x+ 6= 3x 4x = 11 4 + 3= 3 x x x x x = 4 Page 13 of 16
A graph of y= ax + bx+ c given. Use this graph to solve ax bx c + + = 0 1.. 3. Solve using the ZERO or INTERSECT feature of your graphing calculator. Round your answer(s) to the nearest thousandth. Adjust your viewing window as appropriate. 4. 36x 18x 1 0 + + = 5. 1+ 0x = 5x Page 14 of 16
Given the quadratic equation The Quadratic Formula is give by: ax bx c Solve using the Quadratic Formula: + + = 0, we can use the Quadratic Formula to solve the equation for x. ± b b 4 ac x= a 1. x 8x = 16. x 5x = 50 3. x + x+ 3= 0 The Discriminant: b 4ac Page 15 of 16
Section 7.6: Complex Numbers Solve x + = 1 0 imaginary unit, i Complex Number: Imaginary Number: Simplify: 1. 8. 0+ 50 10 3. 5 4 4. 6+ 45 1 Page 16 of 16