Factoring Polynomials: HOW-TO

Factoring Polynomials: HOW-TO

Step 1: Factor out Greatest Common Factor (GCF) using Distributive Property - ALWAYS CHECK FOR GCF

Factor this problem.
Example 1: 2a - 16b = 2 (a - 8b)

Thinking:
1) The GCF of 2 and 16 is 2. There are no variables in common so 2 is the GCF.
2) Write down the 2 and think 2a divided by 2 is a and -16b divided by 2 is -8b. Answer is 2(a - 8b)

Factor this problem.
Example 2: 4x³ - 8x = 4x(x² - 2)

Thinking:
1) The GCF of 4 and 8 is 2. The GCF of x³and x is x. (It will always be the lowest exponent of the variables.)
2) Write down the 4x and think 4x³ divided by 4x is x² and -8x divided by 4x is -2. Answer is 4x(x² - 2)

Step 2: Determine the number of terms and then use appropriate pattern to factor.

4 Terms: What pattern do you see
Factoring a Multivariate Polynomial by Grouping: Problem Type 1 and 2: (6.10, 6.11 in Worktext)

Example 3: 2x + 2y + ax + ay
= 2(x + y) + a(x + y)..Factor each pair (see step 1)
= (x + y)(2 + a).........Factor out (x + y) - again step 1

This last step is "tricky." To help you see it better think of x + y = w. Then the problem is 2w + aw. Now it is easy to see that you are factoring out a w and getting w(2+a). Since w = x + y, the answer is (x + y)(2 + a).

Example 4: xy - 7x - 6y + 42
= x(y - 7) - 6(y - 7)...Factor each pair (see step 1)
= (y - 7)(x - 6)..........Factor out (y - 7)

In this problem the second step is "tricky." To get the signs the same in the (y - 7) factor you had to factor out a "- 6" instead of just 6. So you need to look ahead to be sure that you have two factors the same when you are done with the second step.

Factor these problems.
Example 5: xy - bx - ay + ab
   = x(y - b) - a(y - b)
   = (x - a) (y - b)

Example 6: 2x² + 2ax + 5x + 5a
  = 2x( x + a) + 5( x + a)
= (2x + 5) ( x + a)

2 Terms...Difference of Two Squares or Sum and Difference of Two Cubes
Click here to go to a table of squares and cubes to help you do these problems.

Difference of Two Squares...you are looking for a subtract sign between two quantities which are squares. NOTE: You can NOT factor the sum of two squares.
Factoring the Difference of Two Squares (6. 18 in Worktext)

The pattern for factoring the difference of two squares is, a²- b² = (a + b)(a - b).

Factor this problem.
Example 7: x² - 81 = (x + 9)(x - 9)

Thinking:
1) There is no GCF (Step 1).
2) Has 2 terms, a minus sign, and both terms are squares.
3) Think -> the square root of x²is x and the square root of 81 is 9
4) Write down (x + 9)(x - 9).

Factor this problem.
Example 8:. 2x³ - 32x = 2x(x² - 16) = 2x(x + 4)(x - 4)
     Notice that in this problem your first step was to factor out the GCF of 2x. The resulting factor was the difference of two squares.

Difference of Two Fourth Powers...this is a special case of the difference of two squares.
Factoring the Difference of Two Fourth Powers (6.19 in Worktext)

The pattern for factoring the difference of two fourth powers is, a⁴- b⁴ = (a²+ b² )(a²- b² )=( a²+ b²)(a + b)(a - b).

Factor these problems.
Example 9: x⁴ - 16 = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2)
Do you see that in the first step you are really thinking "difference of two squares?" Then in the second step you cannot factor the plus case of the squares but you can factor x² - 4.

Example 10:. 3yx⁴ - 243y = 3y(x⁴ - 81) = 3y(x² + 9)(x² - 9)=3y (x² + 9)(x + 3)(x - 3)
Notice that in this problem your first step was to factor out the GCF of 3y.

Sum or Difference of Two Cubes...you are looking for two cubes. The sign between them can be addition or subtraction.
Factoring the Sum and Difference of Two Cubes (6.20 in Worktext)

The pattern for factoring the difference of two cubes is

The pattern for factoring the sum of two cubes is

Factor this problem.
Example 11:. 8x³ - 27 = (2x - 3)(4x² + 6x + 9)

Thinking:
1) There is no GCF (Step 1).
2) Has 2 terms, a minus sign, and both terms are cubes.
3) Think -> the cube root of 8x³ is 2x and the cube root of -27 is - 3.
4) Write down (2x - 3)
5) For the second factor, square the 2x getting 4x², multiply the (2x)( - 3) and change the sign giving + 6x, and square the ( - 3) getting + 9.
6) Putting step 4 and 5 together you have (2x - 3)(4x² + 6x + 9)

Factor this problem.
Example 12:. 64x³ + 1 = (4x + 1)(16x² - 4x + 1)

3 Terms...Quadratic Trinomial
NOTE: You can use this strategy for Factoring a Quadratic Trinomial with Leading Coefficient 1 (6.13 in Worktext), Factoring a Quadratic Trinomial with Leading Coefficient Greater than 1 (6.14 in Worktext), Factoring a Perfect Square (6.15 in Worktext), and Factoring the Product of a Quadratic Trinomial with a Monomial (6.17 in Worktext).

type 1: coefficient of x²=1
example: x² + 8x + 12
List the factors of 12: (1, 12),(-1,-12),(2,6),(-2,-6),(3,4),(-3,-4).
Pick the pairs whose sum is 8: (2, 6)
Answer: x² + 8x + 12 = (x + 2)(x + 6)

type 2: coefficients of x² not equal to 1
example: 2x² + 5x - 3
Multiply (2)( - 3). List the factors of - 6: (1,-6),( -1,6),(2,-3)(-2,3)
Find the pair whose sum is 5
Replace the 5x with -1x and 6x
= 2x² - 1x + 6x - 3
Now follow the pattern for factoring with 4 terms
= x(2x - 1) + 3(2x - 1)
= (2x - 1)( x + 3)

Factor these problems.
Example 13: 6x² + 19x + 10
    There is no GCF (see Step 1 above).
    Multiply the (6)(10) = 60 and list all factors that are reasonable: (6)(10), (-6)(-10), (5, 12), (4, 15)
            The sum of (4, 15) is 19 so that is what we want.
    Rewrite 6 x² + 19x + 10 as

              6x² + 4x + 15x + 10 = 2x(3x + 2) + 5(3x + 2) = (3x + 2)(2x + 5).

Example 14: 4x³ - 18x² - 10x
The GCF is 2x. Factor it out. 2x(2x²- 9x - 5)
   Working just with the (2x²- 9x - 5), multiply the (2)(-5) = -10 and list all factors that are reasonable:
   (1)(-10), (-1, 10)
   The sum of (1)(-10) is -9 so that is what we want.
    Rewrite 2x²- 9x - 5 as

              2x² + x - 10x - 5 = x(2x + 1) - 5(2x + 1) = (2x + 1)(x - 5).
   NOW, get the GCF times this part of the answer for the final answer: 2x(2x + 1)(x - 5).