Download 5-5 Dividing Polynomials

5-5A Dividing Polynomials
Name__________________
Objective: To divide polynomials using long division and synthetic division.
Algebra 2 Standard 3.0
*When you divide a polynomial f(x) by a divisor d(x),
you get a quotient polynomial q(x) and a remainder polynomial r(x).
f  x
r  x
 q  x 
d  x
d  x
*Polynomial Long Division: When dividing polynomials, set it up just as you
would with numbers 1  2  means 2 (divisor) 1.0 (dividend) .
* How to set up polynomial division:
1) Write the dividend and divisor in descending powers of the variable.
2) Insert placeholders with zero coefficients for missing powers of the variable.
Ex. 1: Divide f(x) = x3 + 3x2 – 7 by x2 – x – 2 (Notice f(x) is missing x-term, use 0x)
Ex. 2: Divide f(x) = 3x3 + 17x2 + 21x – 11 by x + 3
Alg 2 Advanced Ch.5B Notes page 1
You Try: Divide using long division.
a. (2x4 + x3 + x – 1) ÷ (x2 + 2x – 1)
b. (x3 – x2 + 4x – 10) ÷ (x + 2)
*Synthetic Division can be used to divide any polynomial by a divisor in the form x – k.
Ex. 3: Divide f(x) = 2x3 + 9x2 + 14x + 5 by (x – 3)
You Try: Use Synthetic Division to divide.
a.  x3  4 x 2  x  1   x  3
Alg 2 Advanced Ch.5B Notes page 2
b.
 4x
3

 x 2  3x   x  1
5-5B Apply the Remainder and Factor Theorems
Name__________________
Objective: To divide polynomials using long division and synthetic division.
Algebra 2 Standard 3.0
*Remainder Theorem:
If a polynomial f(x) is divided by (x – k), then the remainder is r  f (k )
Ex.1: Show f (3) is the same number as the remainder when f ( x)  3x3  2 x 2  10 x  6 is
divided by x  3 .
*Factor Theorem:
A polynomial f(x) has a factor (x – k) if and only if f (k )  0
Ex. 2: Factor 2 x3  11x 2  3x  36 completely given that x  3 is a factor.
Ex. 3: A zero of f ( x)  x3  4 x 2  15 x  18 is x  1 . Find the other zeros.
Alg 2 Advanced Ch.5B Notes page 3
You Try: a. Factor 3x3  8 x 2  3x  2
given that x  2 is a factor.
b. A zero of f ( x)  x3  4 x 2  11x  30 is
x  2 . Find the other zeros.
Ex. 4: Show ( x  2) and ( x  5) are factors of f ( x)  x 4  14 x3  69 x 2  140 x  100 .
And then find the other factors.
Alg 2 Advanced Ch.5B Notes page 4
5-6 Find Rational Zeros
Objective: To find all real zeros of a polynomials functions.
Algebra 2 Standard 3.0
* THE RATIONAL ZERO THEOREM
If f(x) = anxn + …+ a1x + a0 has _________ coefficients,
then every rational zero of f has the following form:
p
factor of constant term ( a0 )

q
factor of leading coefficient ( an )
Ex. 1: List all possible zeros of f ( x)  3x3  9 x 2  x  10 .
You Try: List all possible zeros of f ( x)  2 x3  7 x 2  4 x  8 .
Ex. 2: Find zeros when the leading coefficient is 1.
Find all real zeros of f(x) = x3  4x2  7x + 10.
Step 1) List the possible rational zeros.
2) Test these zeros using synthetic division.
3) Factor the trinomial and use the factor theorem.
You Try: Find all real zeros of f(x) = x3 + 3x2  l0x  24
Alg 2 Advanced Ch.5B Notes page 5
Name__________________
Ex. 3: Find zeros when the leading coefficient is not 1.
Find all real zeros of f(x) = 8x4 + 2x3  21x2  7x + 3.
Step 1) List the possible rational zeros.
2) Choose two reasonable values using a graphing calculator.
3) Find the remaining zeros by using synthetic division and quadratic formula.
You Try: Find all real zeros of f(x) = 9x4 + 12x3  26x2  11x + 6.
(You may use a graphing calculator but need to show work.)
Ex. 4: You are building a wooden square sandbox for a local playground. You want the volume
of the box to be 16 cubic feet. You want the height of the box to be x feet and the length of each
side of the square base to be x + 3 feet. What are the dimensions?
Alg 2 Advanced Ch.5B Notes page 6
5-7 Apply the Fundamental Theorem of Algebra
Objective: To classify the zeros of polynomial functions.
Mathematical Analysis 4.0
Name__________________
* Repeated Solution:
For f(x) = 0, k is a repeated solution if and only if the factor (x-k) has a degree greater than 1
when f is factored completely.
Ex)
*THE FUNDAMENTAL THEOREM OF ALGEBRA
Theorem: If f(x) is a polynomial of degree n where n _____ 0, then the equation f(x) = 0 has
at least __________ solution in the set of complex numbers.
Corollary: If f(x) is a polynomial of degree n, then the equation f(x) = 0 has
exactly _____ solutions provided each solution repeated twice is counted as _____ solutions,
each solution repeated three times is counted as _____ solutions and so on.
Ex. 1: Find the number of solutions or zeros for each equation or function.
a) x3  3x2 + 9x  27 = 0
b) f(x) = x4 + 6x3  32x
You Try: State the number of zeros of f(x) = x5  2x3  9x + 18.
Ex. 2: Find all zeros of f(x) = x5  5x4 + 9x3  5x2  8x + 12.
You Try: Find all zeros of f(x) = x4  7x3 + 13x2 + x  20
Alg 2 Advanced Ch.5B Notes page 7
*Complex Conjugates Theorem
If f is a polynomial function with __________ coefficients, and __________ is an imaginary
zero of f, then _______________ also a zero of f.
*Irrational Conjugates Theorem
Suppose f is a polynomial function with __________ coefficients, and __________ is a real
zero of f, then __________ is also a zero of f.
*Sum and Product Theory
If a and b are zeros of f(x), then f ( x)  ( x  a)( x  b)  x 2  (a  b) x  ab .
Ex)
Ex. 2: Write a polynomial function f of least degree that has real coefficients, a leading
coefficient of 1, and 0 , 5 and -2 as zeros.
Ex. 3: Write a polynomial function f of least degree that has real coefficients, a leading
coefficient of 1, and 2 and 3 + i as zeros.
(Because the coefficients are real and 3 + i is a zero, __________ must also be a zero.)
Alg 2 Advanced Ch.5B Notes page 8
You Try: Write a polynomial of least degree that has rational coefficients, a leading coefficient
of 1, and 4 and 1  6 as zeros.
*Descartes' Rule of Signs
Let f(x) = anxn + an 1xn  1 + …….+ a2x2 + a1x + a0
be a polynomial function with real coefficients.
 The number of __________ real zeros of f is equal to the number of changes in sign of
the coefficients of __________ or is less than this by an __________ number.

The number of __________ real zeros of f is equal to the number of changes in sign of the
coefficients of __________ or is less than this by an __________ number
Ex. 4: Determine the possible numbers of positive real zeros, negative real zeros, and imaginary
zeros for f(x) = 2x5  7x4 + 12x3 + 2x2 + 4x + 6.
(+) real (-) real Imaginary
zeros zeros
zeros
You Try: Determine the possible numbers of positive real zeros, negative real zeros, and
imaginary zeros for f(x) = 3x5  4x4 + x3 + 6x2 + 7x  8.
Alg 2 Advanced Ch.5B Notes page 9
Total
zeros