FACTORING POLYNOMIALS

FIRST: COUNT THE TERMS OF THE POLYNOMIAL ,

SECOND: FROM ALL THE TERMS, FACTOR OUT THE LARGEST COMMON
FACTOR,

THIRD: DO YOU HAVE ONE OF THE FOL LOWING , BELOW:

If TWO Terms , do you have:
Difference of two Squares
a² - b² = (a - b)(a + b)

Sum of two Squares
a² + b² = a² + b² (NOT FACTORABLE in R!)

Difference of two Cubes
a³ – b³ = (a - b)(a² + ab + b²)

Sum of two Cubes
a³ + b³ = (a + b)(a² - ab + b²)

If THREE Terms, is it:
Perfect Square Trinomial (PST)
x² + 2xy + y² = (x + y) ² ← square of a sum
x² - 2xy + y² = (x - y) ² ← square of a difference
 

Any Trinomial (over for the product/ sum method )
ax² + bx + c = ( _x + _y )( _x +_y )

If FOUR Terms, try:
Factoring by Grouping (into pairs)
xy - y + x² - x (4 terms)
y(x - 1) + x(x - 1) (2 terms)
(x - 1 )(y + x) (1 term)

If MORE Terms:
Use long division for polynomials.
When the remainder is zero ,
then the divisor is a factor.

THE PRODUCT/SUM METHOD
for factoring any trinomial:
(1st coeff.) x² + (middle coeff.) xy + (last coeff.) y²
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The METHOD:

1) Multiply the first and last coefficients of the trinomial : (1^st) • (last) = ab
2) Rewrite middle coefficient as a sum: (middle) = a + b
3) Find a and b to satisfy equations in #1 and #2. If you cannot find such a and b
then the trinomial is not factorable .
4) Rewrite the middle term as two terms using a and b as their coefficients.
5) The trinomial can now be rewritten with 4 terms.
6) Factor the four terms by grouping. (The end!)

EXAMPLE 1: 9x² - 6xy + 4y² (in standard form)

1) Product of (1^st) • (last) = (4) • (9) = 36, so ab = 36.
2) Middle term = -6, so a + b = -6
3) Find a and b that satisfy #1 and #2. There is no such a and b so the trinomial is
not factorable. (The end!)

EXAMPLE 2: 9x² - 12xy + 4y² (in standard form)
1) Product of (1^st) • (last) = (4) • (9) = 36, so ab = 36.
2) Middle term = -12, so a + b = -12.
3) Find a and b to satisfy #1 and #2: Here a = -6 and b = -6. (Check it!)
4) Rewrite the middle term using the a and the b found in #3: -12xy = -6xy + -6xy
5) Rewrite the trinomial: 9x² - 6xy - 6xy + 4y² = (4 terms)
6) Factor by grouping:
3x(3x - 2y) - 2y(3x - 2y) = (2 terms)
(3x - 2y)(3x - 2y) = (1 term)
(3x - 2y) ²

The end!
(You may have noticed that Example 2 is a perfect square trinomial, a PST.)

OSU - N MATH LAB Prepared by Rose Hart