1 / 32

2.3: Polynomial Division

2.3: Polynomial Division. Objectives: To divide polynomials using long and synthetic division To apply the Factor and Remainder Theorems to find real zeros of polynomial functions. Vocabulary.

mirari
Download Presentation

2.3: Polynomial Division

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 2.3: Polynomial Division Objectives: To divide polynomials using long and synthetic division To apply the Factor and Remainder Theorems to find real zeros of polynomial functions

  2. Vocabulary As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook.

  3. Exercise 1 - - - Use long division to divide 5 into 3462.

  4. Exercise 1 Quotient Divisor Dividend - - - Remainder Use long division to divide 5 into 3462.

  5. Exercise 1 Dividend Remainder Divisor Divisor Quotient Use long division to divide 5 into 3462.

  6. Remainders If you are lucky enough to get a remainder of zero when dividing, then the divisor divides evenlyinto the dividend. This means that the divisor is a factorof the dividend. For example, when dividing 3 into 192, the remainder is 0. Therefore, 3 is a factor of 192.

  7. Dividing Polynomials Dividing polynomials works just like long division. In fact, it is called long division! Before you start dividing: Make sure the divisor and dividend are in standard form (highest to lowest powers). If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder.

  8. Exercise 2 How many times does x go into x2? Multiply x by x + 1 - - Multiply 2 by x + 1 - - Divide x + 1 into x2 + 3x + 5 Line up the first term of the quotient with the term of the dividend with the same degree.

  9. Exercise 2 Quotient Dividend - - - - Divisor Remainder Divide x + 1 into x2 + 3x + 5

  10. Exercise 2 Dividend Remainder Divisor Quotient Divisor Divide x + 1 into x2 + 3x + 5

  11. Exercise 3 Divide 6x3 – 16x2 + 17x – 6 by 3x – 2

  12. Exercise 4 Use long division to divide x4 – 10x2 + 2x + 3 by x – 3

  13. Synthetic Division When your divisor is of the form x - k, where k is a constant, then you can perform the division quicker and easier using just the coefficients of the dividend. This is called fake division. I mean, synthetic division.

  14. Synthetic Division = Add terms k a b c d ka = Multiply by k a Remainder Coefficients of Quotient (in decreasing order) Synthetic Division (of a Cubic Polynomial) To divide ax3 + bx2 + cx + d by x – k, use the following pattern.

  15. Synthetic Division = Add terms k a b c d ka = Multiply by k a Synthetic Division(of a Cubic Polynomial) To divide ax3 + bx2 + cx + d by x – k, use the following pattern. Important Note: You are always adding columns using synthetic division, whereas you subtracted columns in long division.

  16. Synthetic Division = Add terms k a b c d ka = Multiply by k a Synthetic Division (of a Cubic Polynomial) To divide ax3 + bx2 + cx + d by x – k, use the following pattern. Important Note: k can be positive or negative. If you divide by x + 2, then k = -2 because x + 2 = x – (-2).

  17. Synthetic Division = Add terms k a b c d ka = Multiply by k a Synthetic Division (of a Cubic Polynomial) To divide ax3 + bx2 + cx + d by x – k, use the following pattern. Important Note: Add a coefficient of zero for any missing terms!

  18. Exercise 5 Use synthetic division to divide x4 – 10x2 + 2x + 3 by x – 3

  19. Exercise 6 Evaluate f (3) for f (x) = x4 – 10x2 + 2x + 3.

  20. Remainder Theorem If a polynomial f (x) is divided by x – k, the remainder is r = f (k). This means that you could use synthetic division to evaluate f (5) or f (-2). Your answer will be the remainder.

  21. Exercise 7 Divide 2x3 + 9x2 + 4x + 5 by x + 3 using synthetic division.

  22. Exercise 8 Use synthetic division to divide f(x) = 2x3 – 11x2 + 3x + 36 by x – 3. Since the remainder is zero when dividing f(x) by x – 3, we can write: This means that x – 3 is a factorof f(x).

  23. Factor Theorem A polynomial f(x) has a factor x – k if and only if f(k) = 0. This theorem can be used to help factor/solve a polynomial function if you already know one of the factors.

  24. Exercise 9 Factor f(x) = 2x3 – 11x2 + 3x + 36 given that x – 3 is one factor of f(x). Then find the zeros of f(x).

  25. Exercise 10 Given that x – 4 is a factor of x3 – 6x2 + 5x + 12, rewrite x3 – 6x2 + 5x + 12 as a product of two polynomials.

  26. Exercise 11 Find the other zeros of f(x) = 10x3 – 81x2 + 71x + 42 given that f(7) = 0.

  27. Rational Zero Test: we use this to find the rational zeros for a polynomial f(x). It says that if f(x) is a polynomial of the form: Then the rational zeros of f(x) will be of the form: Where p = factor of the constant & q = factor of leading coefficient Rational zero = Possible rational zeros = factors of the constant term___ factors of the leading coefficient • Keep in mind that a polynomial can have rational zeros, irrational zeros and complex zeros.

  28. Ex 1: Find all of the possible rational zeros of f(x)

  29. Ex 2: Find the rational zeros of: Let’s start by listing all of the possible rational zeros, then we will use synthetic division to test out the zeros: 1. Start with a list of factors of -6 (the constant term): p = 2. Next create a list of factors of 1 (leading coefficient): q = 3. Now list your possible rational zeros: p/q = Testing all of those possibilities could take a while so let’s use the graph of f(x) to locate good possibilities for zeros. Use your trace button!

  30. Ex 2 continued: Find all of the rational zeros of the function

  31. Ex 3: Find all the real zeros of : p = Factors of 3: q = Factors of 2: Candidates for rational zeros: p/q = Let’s look at the graph: Which looks worth trying? Now use synthetic division to test them out.

  32. Homework • Dividing Polynomials Worksheet Page 127-128 36,38, 49-59 odd

More Related