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5-5A Dividing Polynomials Name__________________ Objective: To divide polynomials using long division and synthetic division. Algebra 2 Standard 3.0 *When you divide a polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) and a remainder polynomial r(x). f x r x q x d x d x *Polynomial Long Division: When dividing polynomials, set it up just as you would with numbers 1 2 means 2 (divisor) 1.0 (dividend) . * How to set up polynomial division: 1) Write the dividend and divisor in descending powers of the variable. 2) Insert placeholders with zero coefficients for missing powers of the variable. Ex. 1: Divide f(x) = x3 + 3x2 – 7 by x2 – x – 2 (Notice f(x) is missing x-term, use 0x) Ex. 2: Divide f(x) = 3x3 + 17x2 + 21x – 11 by x + 3 Alg 2 Advanced Ch.5B Notes page 1 You Try: Divide using long division. a. (2x4 + x3 + x – 1) ÷ (x2 + 2x – 1) b. (x3 – x2 + 4x – 10) ÷ (x + 2) *Synthetic Division can be used to divide any polynomial by a divisor in the form x – k. Ex. 3: Divide f(x) = 2x3 + 9x2 + 14x + 5 by (x – 3) You Try: Use Synthetic Division to divide. a. x3 4 x 2 x 1 x 3 Alg 2 Advanced Ch.5B Notes page 2 b. 4x 3 x 2 3x x 1 5-5B Apply the Remainder and Factor Theorems Name__________________ Objective: To divide polynomials using long division and synthetic division. Algebra 2 Standard 3.0 *Remainder Theorem: If a polynomial f(x) is divided by (x – k), then the remainder is r f (k ) Ex.1: Show f (3) is the same number as the remainder when f ( x) 3x3 2 x 2 10 x 6 is divided by x 3 . *Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f (k ) 0 Ex. 2: Factor 2 x3 11x 2 3x 36 completely given that x 3 is a factor. Ex. 3: A zero of f ( x) x3 4 x 2 15 x 18 is x 1 . Find the other zeros. Alg 2 Advanced Ch.5B Notes page 3 You Try: a. Factor 3x3 8 x 2 3x 2 given that x 2 is a factor. b. A zero of f ( x) x3 4 x 2 11x 30 is x 2 . Find the other zeros. Ex. 4: Show ( x 2) and ( x 5) are factors of f ( x) x 4 14 x3 69 x 2 140 x 100 . And then find the other factors. Alg 2 Advanced Ch.5B Notes page 4 5-6 Find Rational Zeros Objective: To find all real zeros of a polynomials functions. Algebra 2 Standard 3.0 * THE RATIONAL ZERO THEOREM If f(x) = anxn + …+ a1x + a0 has _________ coefficients, then every rational zero of f has the following form: p factor of constant term ( a0 ) q factor of leading coefficient ( an ) Ex. 1: List all possible zeros of f ( x) 3x3 9 x 2 x 10 . You Try: List all possible zeros of f ( x) 2 x3 7 x 2 4 x 8 . Ex. 2: Find zeros when the leading coefficient is 1. Find all real zeros of f(x) = x3 4x2 7x + 10. Step 1) List the possible rational zeros. 2) Test these zeros using synthetic division. 3) Factor the trinomial and use the factor theorem. You Try: Find all real zeros of f(x) = x3 + 3x2 l0x 24 Alg 2 Advanced Ch.5B Notes page 5 Name__________________ Ex. 3: Find zeros when the leading coefficient is not 1. Find all real zeros of f(x) = 8x4 + 2x3 21x2 7x + 3. Step 1) List the possible rational zeros. 2) Choose two reasonable values using a graphing calculator. 3) Find the remaining zeros by using synthetic division and quadratic formula. You Try: Find all real zeros of f(x) = 9x4 + 12x3 26x2 11x + 6. (You may use a graphing calculator but need to show work.) Ex. 4: You are building a wooden square sandbox for a local playground. You want the volume of the box to be 16 cubic feet. You want the height of the box to be x feet and the length of each side of the square base to be x + 3 feet. What are the dimensions? Alg 2 Advanced Ch.5B Notes page 6 5-7 Apply the Fundamental Theorem of Algebra Objective: To classify the zeros of polynomial functions. Mathematical Analysis 4.0 Name__________________ * Repeated Solution: For f(x) = 0, k is a repeated solution if and only if the factor (x-k) has a degree greater than 1 when f is factored completely. Ex) *THE FUNDAMENTAL THEOREM OF ALGEBRA Theorem: If f(x) is a polynomial of degree n where n _____ 0, then the equation f(x) = 0 has at least __________ solution in the set of complex numbers. Corollary: If f(x) is a polynomial of degree n, then the equation f(x) = 0 has exactly _____ solutions provided each solution repeated twice is counted as _____ solutions, each solution repeated three times is counted as _____ solutions and so on. Ex. 1: Find the number of solutions or zeros for each equation or function. a) x3 3x2 + 9x 27 = 0 b) f(x) = x4 + 6x3 32x You Try: State the number of zeros of f(x) = x5 2x3 9x + 18. Ex. 2: Find all zeros of f(x) = x5 5x4 + 9x3 5x2 8x + 12. You Try: Find all zeros of f(x) = x4 7x3 + 13x2 + x 20 Alg 2 Advanced Ch.5B Notes page 7 *Complex Conjugates Theorem If f is a polynomial function with __________ coefficients, and __________ is an imaginary zero of f, then _______________ also a zero of f. *Irrational Conjugates Theorem Suppose f is a polynomial function with __________ coefficients, and __________ is a real zero of f, then __________ is also a zero of f. *Sum and Product Theory If a and b are zeros of f(x), then f ( x) ( x a)( x b) x 2 (a b) x ab . Ex) Ex. 2: Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 0 , 5 and -2 as zeros. Ex. 3: Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 2 and 3 + i as zeros. (Because the coefficients are real and 3 + i is a zero, __________ must also be a zero.) Alg 2 Advanced Ch.5B Notes page 8 You Try: Write a polynomial of least degree that has rational coefficients, a leading coefficient of 1, and 4 and 1 6 as zeros. *Descartes' Rule of Signs Let f(x) = anxn + an 1xn 1 + …….+ a2x2 + a1x + a0 be a polynomial function with real coefficients. The number of __________ real zeros of f is equal to the number of changes in sign of the coefficients of __________ or is less than this by an __________ number. The number of __________ real zeros of f is equal to the number of changes in sign of the coefficients of __________ or is less than this by an __________ number Ex. 4: Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f(x) = 2x5 7x4 + 12x3 + 2x2 + 4x + 6. (+) real (-) real Imaginary zeros zeros zeros You Try: Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f(x) = 3x5 4x4 + x3 + 6x2 + 7x 8. Alg 2 Advanced Ch.5B Notes page 9 Total zeros