Algebra 6.4 Special Factoring Techniques

Front 1

Factoring a difference of squares

Back 1

x2 - y2 = (x + y) (x - y)

1. Both terms of the binomial must be squares such as
x2, 9y2, 25, 1, m4

2. the second terms of the binomials must have different signs ( one positive & one negative)

Front 2

factoring a difference os squares examples

Back 2

1. a2 - 49 = (a + 7) (a -7)

2. y2 - m2 = (y + m) (y - m)

3. z2 - 9/16 = (z + 3/4) (z - 3/4)

4. x2 - 8 = 8 is not a square of an integer, this binomial is not a difference of squares. thus a prime polynomial.

5. p2 + 16 = is a sum of squares is not equal to (p + 4) (p - 4), and has no common factors to factor out, thus cannot be factored and a prime polynomial.

Front 3

Caution 1

Back 3

After any common factor is removed, a sum of squares cannot be factored.

Front 4

Caution 2

Back 4

Factor again when any of the factors is a difference of squares

= m4 - 16
= (m2)2 - (4)2
= (m2 = 4) (m2 - 4)
= (m2 + 4) (m + 4) (m - 4)

Front 5

Note 1

Back 5

Always check factored form by multiplying.

= m2 - 16
= m2 - (4)2
= (m + 4) (m - 4)

check:
= m2 - 4m + 4m - 16
= m2 -16

Front 6

Perfect Square Trinomial

Back 6

is a trinomial that is a square of a binomial; for example,

x2 + 8x + 16 is a perfect square trinomial because it is the square of the binomial
x + 4.

= x2 + 8x + 16
= (x + 4) ( x - 4)
= (x + 4)2

Front 7

Factoring Perfect Square Trinomials

Back 7

x2 + 2xy + y2 = (x + y)2

x2 - 2xy + y2 = (x - y)2

Front 8

The middle term of a Perfect Square Trinomial is........

Back 8

always twice the product of the two terms in the squared binomial. Use this rule to check any attempt to factor a trinomial that appears to be a perfect square.

Front 9

Note 2

Back 9

1. the sign of the second term in the squared binomial is always the same as the sign of the middle term in the trinomial.

2. The first and last terms of a perfect square trinomial must be positive because they are squares. For example the polynomial x2 - 2x - 1 cannot be a perfect square, because the last term is negative.

3. Perfect square trinomials can also be factored by using grouping or the FOIL method.

Front 10

Factoring a difference of Cubes

Back 10

x3 - y3 = (x - y) (x2 + xy = y2)

this pattern for factoring a difference of cubes should be memorized.

Front 11

Factoring a difference of Cubes
example:

= x3 - y3

= (x - y) ( x2 + xy + y2)

Back 11

Notice the pattern of the terms in the factored form of x3 - y3:

1. x3 - y3 = (a binomial factor) ( a trinomial factor)

2. the binomial factor has the difference of the cube roots of the given terms.

3. The terms in the trinomial factor are all positive.

4. What you write in the binomial factor determines the trinomial factor.

Front 12

Caution 3

Back 12

The polynomial x3 - y3 is not equivalent to (x -y )3 , because (x - y)3 can also be written:

(x - y)3 = (x - y) ( x - y) ( x - y)
= (x - y) (x2 - 2xy + y2)

but,

=x3 - y3
= (x - y) ( x2 + xy + y2)

Front 13

Factoring a Sum of Cubes

Back 13

= x3 + y3
= (x + y) (x2 - xy + y2)

Front 14

Difference of Squares Formula

Back 14

= x2 - y2
= (x + y) ( x - y)

Front 15

Perfect Square Trinomials Formula

Back 15

= x2 + 2xy + y2
= ( x + y )2

=x2 - 2xy + y2
= ( x - y )2

Front 16

Difference of Cubes Formula

Back 16

= x3 - y3
= (x - y) ( x2 + xy + y2)

Front 17

Sum of Cubes

Back 17

= x3 + y3
= (x + y) ( x2 - xy + y2)