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Solving Quadratic Equations by Completing the Square
Graphing Logarithmic Functions
Division Property of Exponents
Adding and Subtracting Rational Expressions With Like Denominators
Rationalizing the Denominator
Multiplying Special Polynomials
Functions
Solving Linear Systems of Equations by Elimination
Solving Systems of Equation by Substitution and Elimination
Polynomial Equations
Solving Linear Systems of Equations by Graphing
Quadratic Functions
Solving Proportions
Parallel and Perpendicular Lines
Simplifying Square Roots
Simplifying Fractions
Adding and Subtracting Fractions
Adding and Subtracting Fractions
Solving Linear Equations
Inequalities in one Variable
Recognizing Polynomial Equations from their Graphs
Scientific Notation
Factoring a Sum or Difference of Two Cubes
Solving Nonlinear Equations by Substitution
Solving Systems of Linear Inequalities
Arithmetics with Decimals
Finding the Equation of an Inverse Function
Plotting Points in the Coordinate Plane
The Product of the Roots of a Quadratic
Powers
Solving Quadratic Equations by Completing the Square

Factoring Polynomials

A strategy for factoring polynomials is given in the following box.

Strategy for Factoring Polynomials

1. If there are any common factors, factor them out first.

2. When factoring a binomial, look for the special cases: difference of two squares, difference of two cubes, and sum of two cubes. Remember that a sum of two squares a2 + b2 is prime.

3. When factoring a trinomial, check to see whether it is a perfect square trinomial.

4. When factoring a trinomial that is not a perfect square, use grouping or trial and error.

5. When factoring a polynomial of high degree, use substitution to get a polynomial of degree 2 or 3, or use trial and error.

6. If the polynomial has four terms, try factoring by grouping. 

 

Example

Using the factoring strategy

Factor each polynomial completely.

a) 3w3 - 3w2 - 18w

b) 10x2 + 160

c) 16a2b -80ab + 100b

d) aw + mw + az + mz

Solution

a) The greatest common factor (GCF) for the three terms is 3w:

3w3 - 3w2 - 18w = 3w(w2 - w - 6) Factor out 3w.
  = 3w(w - 3)(w + 2) Factor completely.

b) The GCF in 10x2 + 160 is 10:

10x2 + 160 = 10(x2 + 16)

Because x2 + 16 is prime, the polynomial is factored completely.

c) The GCF in 16a2b - 80ab + 100b is 4b:

16a2b - 80ab + 100b = 4b(4a2 - 20a + 25)
  = 4b(2a - 5)2

d) The polynomial has four terms, and we can factor it by grouping:

aw + mw + az + mz = w(a + m) + z(a + m)
  = (w + z)(a + m)

 

Helpful hint

When factoring integers, we write 4 = 2 · 2. However, when factoring polynomials we usually do not factor any of the integers that appear. So we say that 4b(2a - 5)2 is factored completely.

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