1 / 12

2.4a-Zeros of Polynomial Functions

2.4a-Zeros of Polynomial Functions. Sarah Byom and Samantha Kingery. singaporeolevelmaths.com. funnypicss.com. tumblr.com. Remainder Theorem . If a polynomial f(x) is divided by x-k, then the remainder is r=f(k) Example: Find the remainder when f(x)=3x 2 +7x-20 is divided by x-2.

vlora
Download Presentation

2.4a-Zeros of Polynomial Functions

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. 2.4a-Zeros of Polynomial Functions Sarah Byom and Samantha Kingery

  2. singaporeolevelmaths.com funnypicss.com tumblr.com

  3. Remainder Theorem • If a polynomial f(x) is divided by x-k, then the remainder is r=f(k) • Example: Find the remainder when f(x)=3x2+7x-20 is divided by x-2. K=2 (plug 2 in for x), so r=f(2)=3(2)2+7(2)-20=12+14-20=6. The remainder is 6.

  4. Remainder Theorem • Example: Find the remainder when f(x) = x3-x2+4x+10 is divided by x-3 • Answer: Remainder = 40.

  5. Factor Theorem • F(x) has a factor x-k if and only if f(k)=0. • Example: Is x-7 a factor of x2-10x+21? Test: (7)2-10(7)+21 = 0. Yes, the factors are (x-7) and (x-3).

  6. Long Division • Use long division to find the quotient and remainder when 3x3+5x2+8x+7 is divided by 3x+2. 1x2+1x+2 Quotient 3x+2 )3x2+5x2+8x+7 Dividend 3x2+2x2 Multiply: 1x2(3x+2) 3x2+8x+7 Subtract 3x2+2xMultiply: 1x(3x+2) 6x+7 Subtract 6x+4Multiply: 2(3x+2) 3 Remainder

  7. Long Divison • Example: Find the quotient and remainder when x3-x2+4x+10 is divided by x-3. • Answer: x2+2x+10 R: 40

  8. Synthetic Division • Synthetic division is the shortcut method for the division of a polynomial by a linear divisor x-k. • Example: Divide 2x3-3x2-5x-12 by x-3. Zeroof divisor 3 2 -3 -5 -12 Dividend Line for products 6 9 12 Line for sums 2 3 4 0 Remainder Quotient • Multiply the zero of divisor (3) by the first coefficient of the dividend (2). Write the product about the line and one column to the right. • Add the next coefficient of the dividend to the product just found and record the sum below the line in the same column. • Repeat the “multiply” and “add” steps until the last row is complete.

  9. Synthetic Division • Divide 3x3-2x2+x-5 by x-1 using synthetic division. • Answer: 3x2+x+2 R: 3

  10. Finding the Real Zeros of a Polynomial Funciton • Find all of the real zeros of f(x) =2x2-7x3-8x2+14x+8. • Use the Rational Zeros Theorem: Factors of 8 : +1, +2, +4, +8 =+1, +2, +4, +8, +1/2 Factors of 2: +1, +2 • Graph the function, and compare the x-intercepts with the list of zeros. • Then use synthetic division with the possible zeros, which in this case would be 4 and -1/2. 4 2 -7 -8 14 8 -1/2 2 1 -4 -2 8 4 -16 -8 -1 0 2 2 1 -4 -2 0 2 0 -4 0 • Factor (x-4)(x+1/2)(2x2-4)

  11. Finding the Real Zeros of a Polynomial Function • Write the function as a product of linear and irreducible quadratic factors f(x)=x3-x2+x-6 • Answer: (x-2)(x2+x+3)

  12. Works Cited • Demana, Franklin D., Bert K. Waits, Gregory D. Foley, and Daniel Kennedy. Precalculus: Graphical, Numerical, Algebraic. 8th ed. Boston: Pearson Education, Inc. , 2011. Print

More Related