theorems - washington-liberty...aug 03, 2017 · the factor theorem: 1. if p(c)=0, then x-c is a...
TRANSCRIPT
October 26, 2017
3.2 - Synthetic Division and Remainder Theorems
Example 1: Use long division to divide 875 by 3
Key Terms:
Divisor
Dividend
Quotient
Remainder
Dividend
Divisor
Quotient Remainder
This means the answer is written as such:
Divisor
Dividend
October 26, 2017
Division can also be written as:Dividend = (divisor)(quotient)+remainder
What if we wanted to undo the division? What would we multiply both sides by?
If the remainder is 0, then the divisor is a factor of the dividend
October 26, 2017
Example 2: Divide 5x2-4x+17 by x-1. Write your answer as:
a)
b)
October 26, 2017
Example 3: Divide 2x3+4x+7 by x2-1. Write your answer as:
a)Remember to use place-holder zeros for missing values!
b)
October 26, 2017
Synthetic Division-Can only be used if the divisor is a linear factor (x+b). Divide 5x2-4x+17 by x-1
1 5 4 17zero of th
e divisor coefficients of
the dividend
5
Example 4: Use synthetic division to divide -4x3+2x2-11 by x-3
October 26, 2017
b) Is x+1 a factor of P(x)?
Example 5:
a) Find the remainder if p(x)= 2x3+4x2-8x-15 is divided by (x+1) using division
c) Evaluate p(-1). What do you notice?
October 26, 2017
The Remainder TheoremIf a polynomial p(x) is divided by (x-c), then the remainder is equal to p(c).
Example 6: Use the remainder theorem to evaluate p(x)=2x4-5x3+2x2+x-8 for p(-5). Verify with substitution.
October 26, 2017
Example 7: Use the remainder theorem to evaluate the following: f(x)=x4+x3-10x2-4x+24
a) f(-1)
b) f(2)
October 26, 2017
What does a remainder of 0 mean????
*** If the remainder is 0, then (x-c) is a factor of p(x) *** This is called the Factor Theorem
The Factor Theorem:
1. If p(c)=0, then x-c is a factor of p(x)
2. If x-c is a factor of p(x), then p(c)=0.
October 26, 2017
Example 8: Determine if the following are factors of g(x)=x3-13x+12
a) x+1 b) x-2
Example 9: Given that (x+3) is a factor of x3+3x2-4x-12, find the other linear factors.
October 26, 2017
Example 10: Given that (x-1) is a factor of x3-x2-5x+5, then find the other linear factors.
Example 11: It is know that a polynomial is degree 3 with factors (x-2) and (x+⎷5).
a) What is the 3rd linear factor?
b) What is the polynomial in standard form?
October 26, 2017
Example 12: A polynomial is degree 4 with two zeros at 3⎷2 and -⎷6.
a)What are the other two zeros?
b) Write the polynomial as a product of linear factors.
Square roots factors will always come in pairs as conjugates of each other.
(x-⎷a) and (x+⎷a)
October 26, 2017
Example 13: Given that (x+5) is a factor of x3+5x2+4x+20, then find the other linear factors.
Example 14: A polynomial is degree 4 with two zeros at 4, 2, and 3i.
a)What is the other zero?
b) Write the polynomial as a product of linear factors.
October 26, 2017
c) Write the polynomial in standard form.
Example 15: Write x3-8 as a product of linear factors.
October 26, 2017