Warm Up
Factor the polynomials by pulling out the GCF
1) 5x3+20x2
2) 3x+12
Factoring by Grouping
and Perfect Square
Trinomials
What do you notice about the factors of
the two binomials?
5x3+20x2 = 5x2(x+4)
3x+12 = 3(x+4)
They both have a common factor of (x+4)
What do you notice about both of the
binomials on the left side of the equal sign?
The two coefficients have a 1 to 4 ratio
The exponent decreases by 1 from the first
term to the second term
Binomials pairs with coefficients of the same ratio
and variables with exponents that decrease or increase
the same amount, have common binomial factors
What would happen if we combined the two
binomials?
5x3+20x2 = 5x2(x+4)
3x+12 = 3(x+4)
5x3+20x2+3x+12 = 5x2(x+4) + 3(x+4)
GCF
= (x+4)(5x2+3)of (x+4)
thus 5x3+20x2+3x+12 can be factored to
(x+4)(5x2+3)
This is called Factor by Grouping
Determine if the polynomial can be factored by
grouping, and if so, factor by grouping
2x3+4x2+7x+14 yes, 2 to 4 = 7 to 14
and same change of the exponents
3 -> 2 and 1 -> 0
We can factor by grouping by breaking the polynomial
into two binomials, pulling out the GCF from
each binomial, and then pulling out the common
binomial factor
2x3+4x2+7x+14 =
2x3+4x2 + 7x+14 =
2x2(x+2) + 7(x+2) = (x+2)(2x2+7)
thus,
2x3+4x2+7x+14 = (x+2)(2x2+7)
Factor by grouping, if possible
1) 9x3+18x2+7x+14 2) x4+12x3+4x2+48x
3) x3 + 3x2 - 4x - 12 4) 7x4+14x3+2x2+5x
Perfect Square Trinomials
Similar to the way numbers have perfect squares,
2
25 = 5
t
rinomials can have perfect binomial squares. For example
x2+10x+25 = (x+5)(x+5)=(x+5)2
(a=1, b=10, c=25)
For a trinomial with a leading coefficient of 1 (a=1),
2
x +bx+c
the trinomial is a perfect square trinomial if
c= ( )2
wh
en this is the case, the perfect square trinomial factors to
b
2
b2
(x+ )
2
Determine if the trinomials are perfect square trinomials, and if so, give the factors.
1) x2+14x+49 2) x2-8x+64 3) x2 - 6x + 9
What if the leading coefficient is greater than 1?
To determine if it is a
perfect square trinomial,
b
(( 2
= ac
2
1) 4x2 + 20x + 25 2) 9x2 - 12x + 4
Worksheet