Synthetic Substitution
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1 Write your questions and thoughts here! 7.2 Synthetic and Long Polynomial Division 1 Use direct substitution to evaluate t a = a! a! + a + 5 when a = - 2 Synthetic Substitution Synthetic substitution is a method for evaluating a polynomial that uses fewer steps. Step 1: Write the value to be evaluated outside and the coefficients in descending order inside. Step 2: Bring down the leading coefficient and multiply by the number on the left. Step 3: Write the product from the last step under the second coefficient. Add and bring down. Step 4: Multiply the sum from the last step by the number on the left. Step 5: Repeat for remaining coefficients. The final sum is the value of f(x). Synthetic Substitution Example: evaluate t x = x! 7x! + 3x! 2 when x = - 1 t a = a! a! + a + 5 when a = - 2 You try! Evaluate f(x) = 3x 4 2x 3 + 4x 2 6x - 1 at x = - 3. Remember about? (shout out to Mr. Wagneezy!) Polynomial a. Divide f(x) = 3x 4 5x 3 + 4x - 6 by (x 2 3x + 5) b. Divide f(x) = 6m 4 12m 3 + m 2 by (m 2) You Try!: c. Divide n 4 + 3n 3 7n 2 21n by (n + 3)
2 7.2 Synthetic and Long Polynomial Division 2 is a method for dividing polynomials that is quicker and more efficient than long division: Examples: d. Divide f(x) = x 3 + 5x 2 7x + 2 by x 2 e. Determine if (x + 3) is a factor of f(x) = 2x 3 + x 2 8x + 21 by using synthetic division. If so, factor completely. f. Suppose you know that x = - 2 is a zero of the function f(x) = x 3 + 2x 2 9x 18. Find the other zeros. Application! Suppose the profit P (in millions of dollars) for a new Algebros T- shirt manufacturer can be modeled by P = - x 3 + 4x 2 + x where x is the number of Bro- Shirts made (in millions). Currently the company produces 4 million shirts and makes a profit of $4,000,000. Can the company make a lesser number of bro- shirts and still make the same profit?
3 7.2 Synthetic and Long Polynomial Division 3 Practice 7.2 Evaluate each function at the given value using synthetic substitution. 1. g m = m! 10m! + 25m + 2 at m = f x = x! x + 24 at x = r t = 3t! 8t! 11t + 2 at t = g x = 5x! + x! x 41 at x = - 5 Divide each polynomial using both long division and synthetic division. Remember, your answers should match J 5. n! + 3n! 9n 38 n + 3 Is (n + 3) a factor of the function? 6. x! + 16x! + 75x! + 91x + 49 x + 7 Is n = - 7 a zero of the function?
4 7. 4a! 36a! + 60a + 72 a 6 Is a = 6 a zero of the function? 8. b! 4b! + 5b! + 8b 14 b 2 Is b = 2 a zero of the function? Factor each polynomial completely. I m a nice guy, so I ll give you one of the factors. You answer for each should consist of 3 binomials. 9. f x = 5x! 18x! 33x 10 (One factor is x 5.) 10. f x = 25x! 40x! + 17x 2 (One factor is x 1.) Find all the zeros of the given polynomial. I m still a nice guy, so I ll give you one of the zeros. 11. f x = 15x! 28x! + 15x 2 (One zero is x = 1.) 12. f x = 9x! + 3x! 5x + 1 (One zero is x = - 1.)
5 7.2 Synthetic and Long Polynomial Division 5 Application If f x = 6x! + 7x! 18x + 5 and one factor of f x is x 1, completely factor f x. 2. Is m = 7 a zero of f m = m! 8m! + 7m!? Synthetic division clearly simplifies the long division process for dividing by a simple binomial (x b), but is there a way to use synthetic division when dividing by a linear expression of the form (ax b) where a > 1? Have you noticed that every synthetic division problem so far had a divisor with a leading coefficient of Use long division to divide 6x! 11x! 5x + 12 by (2x 3). 4. Use synthetic division to divide 6x! 11x! 5x + 12 by x!! 5. Compare the quotients you calculated in #3 and #4 and the factors 2x 3 and x! that you divided by. Now,! explain how to use synthetic division to divide by a linear expression of the form (ax b) where a > 1. From this point forward, you should be able to divide synthetically, even if the leading coefficient is not a Find the missing dimensions: Hint: Divide the volume by (3x-1); then factor! Volume = 3x 3 x 2 27x + 9? (3x-1)?
6 7.2 Synthetic and Long Polynomial Division 6 GRAPH Below, the graph of f x = x 4! + 4 is sketched in bold. Its parent function f x = x! is represented by the thin curve. 1. Describe the translation of the parent graph. Algebra Skillz SIMPLIFY SOLVE 5. Solve: 4(x 1) 2x 3 = 0 2. How does the translation relate to the equation? 4. 3x! 4x 5x x! 6. Factor and solve: 2x! + 5x 3 = 0 MUTIPLE CHOICE SAT Review! Free Response Determine the number of zeros that are positive integers for the function: What is the remainder when x! 4x! + 4x! 10 is divided by (x 3)? f x = 6x! x! 12x 5 (A) 0 (B) 1 (C) 2 (D) 3 (E) Cannot be determined
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