Download presentation

Presentation is loading. Please wait.

Published byPhebe Short Modified over 6 years ago

1
Factors, Remainders, and Roots, Oh My! 1 November 2010

2
Remainders Is there any way I can figure out my remainder in advance? (3x 4 – 8x 3 + 9x + 5) ÷ (x – 2) 3x 3 – 2x 2 – 4x + 1 Remainder 7

3
Remainder Theorem If a polynomial f(x) is divided by x – c, then the remainder is f(c). Like synthetic division, the divisor must be in the form x – c. If it isn’t, we must alter to the divisor to include subtraction.

4
Remainder Theorem, cont. (3x 4 – 8x 3 + 9x + 5) ÷ (x – 2) f(2) = 3(2) 4 – 8(2) 3 + 9(2) + 5 f(2) = 7

5
Remainder Theorem, cont. (2x 4 + 5x 3 − 2x − 8) ÷ (x + 3) x + 3 x – (-3) f(-3) = 2(-3) 4 + 5(-3) 3 – 2(-3) – 8 f(-3) = 25

6
Your Turn On page 249 in your textbook, complete problems 10 – 16. You will Solve for the quotient using synthetic division Check your remainder using the Remainder Theorem

7
Remainders and Factors If a polynomial f(x) is divided by x – a, and f(a) = 0, then x – a is a factor of the polynomial. The Factor Theorem

8
Remainders and Factors, cont. Similarly, if a divisor has a remainder of zero, than the quotient is also a factor of the polynomial.

9
Remainders and Factors, cont. Ex. (a 4 – 1) ÷ (a – 1) = a 3 + a 2 + a + 1 Both a – 1 and a 3 + a 2 + a + 1 are factors of a 4 – 1!

10
Your Turn: On pg. 249 in your textbook, complete problems 41 – 46. You will: Use the Factor Theorem to determine if the given h(x) is a factor of f(x). Confirm your results using synthetic division.

11
Maximum Number of Roots A polynomial of degree n has at most n different roots. Example: f(x) = x 2 – 3x + 4 has at most 2 different roots 0 = (x – 3)(x – 1); x = 1, 3

12
Maximum Number of Roots, cont. However, a polynomial can have less than the maximum number of different roots. This is because roots can repeat. Example: f(x) = x 2 – 10x + 25 0 = (x – 5)(x – 5); x = 5

13
Other Roots Connections Let f(x) be a polynomial. If r is a real number for which one of the following statements is true, then all of the following statements are true: r is a zero of f(x)

14
Other Roots Connections, cont. r is an x-intercept of f(x) x = r is a solution or root when f(x) = 0 x – r is a factor of the polynomial f(x)

15
Applications We can use the maximum number of roots and the root connections to construct the equation of a polynomial from its graph.

16
Applications, cont. x-intercepts: Zeros: Solutions: Max Degree: Linear Factors:

17
Applications, cont. Linear Factors: (x+1)(x – 3) Equation:

18
Your Turn: On page 249 in your textbook, complete problems 51 – 53. You will: List the x-intercepts List the zeros List the solutions Determine the maximum degree Product of the linear factors Determine the equation of a graph

19
Hmwk: Pg. 317: 1 – 5

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google