5-4A Factor and Solve Polynomial Equations
Name _________________
Objective: To factor other polynominal equations.
Algebra 2 Standard 4.0
*Factoring polynomials: A factorable polynomial with integer coefficients is factored completely
if it is written as a product of unfactorable polynominals with integer coefficients.
Factored completely:
Not factored completely:
Ex. 1: Factor the polynomial completely.
a. y3 – 4y2 – 12y
b. 3x3 + 30x2 + 75x
You Try: Factor the polynomial completely.
a. x3 – 7x2 + 10x
b. 3y5 – 75y3
*Special Factoring Patterns:
Sum of Two Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
Difference of Two Cubes
a3 – b3 = (a – b)(a2 + ab + b2)
Ex.
Ex.
Ex. 2: Factor the polynomial completely.
a. 27x3 + 125
b. -2d5 + 250d2
You Try: Factor the polynomial completely.
a. 8b5 + 343b2
Algebra 2 Ch.5B Notes page 1
b. w3 - 27
c. 5g5 – 80g3
*Factoring by Grouping: For some polynomials, you can factor by grouping pairs of terms
that have a common monomial factor. The pattern is shown below:
ra  rb  sa  sb  r  a  b   s  a  b 
  a  b  r  s 
Ex. 3: Factor the polynomial completely.
27r3 + 45r2 – 3r – 5
You Try: Factor the polynomial completely.
x3 + 7x2 – 9x – 63
*Quadratic Form: An expression in the form au2 + bu + c, where u is any expression in x,
is said to be in Quadratic Form.
Ex. 4: Factor the polynomial completely.
a. 10x4 – 10
b. 3m12 + 54m7 + 51m2
You Try: Factor the polynomial completely.
a. 16g4 – 625
Algebra 2 Ch.5B Notes page 2
b. 4h6 – 20h4 + 24h2
5-4B Factor and Solve Polynomial Equations
Objective: To factor other polynominal equations.
Algebra 2 Standard 4.0
Name _________________
Algebra 2 Notes
*Solving polynomial equations: To solve polynomial equations by factoring,
factor the polynomial and use the zero product property.
Ex.1: What are the real number solutions of each equation?
a) 4x5 – 40x3 + 36x = 0
b) 2x5 = 12x3 – 16x ?
c) -27x3 + 15x2 = -6x4 ?
You Try: What are the real number solutions of each equation?
a) 2x5 + 24x = 14x3
b) 6x3 + x2 = 2x
c) 17x3 + 6x2 = 3x4
Algebra 2 Ch.5B Notes page 3
5-5 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 1 .
 
* 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
Algebra 2 Ch.5B Notes page 4
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
Algebra 2 Ch.5B Notes page 5
b.
 4x
3

 x 2  3x  7   x  1