GPS Algebra
Date: ________
Factoring with Special Products: Cubed Root
Binomials
(a + b)3
Distribute Several Times
Polynomial
(Combine Like Terms)
a3 + 2a2b + 2ab2 + a2b + ab2 + b3
a3 + 3a2b + 3ab2 + b3
x3 + 2x2 + 1x + 1x2 + 2x + 1
x3 + 3x2 + 3x + 1
(a + b)∙(a + b)∙(a + b)
(a + b)(a2 + 2ab + b2)
(x + 1)3
(x + 1)∙(x + 1)∙(x + 1)
(x +
1)(x2
+ 2x + 1)
How many terms are in the product? ____ 4 ____________
The first term in the product is always _____ a3 _____________.
The last term in the product is always _____ b3 _____________.
The second term in the product is divisible by the “b” term and by __ 3 ____.
The third term in the product is divisible by the “b” term squared and by __ 3 ____.
Do the following polynomials fit the rule for factoring by a binomial cubed?
Ex1: x3 + 6x2 + 12x + 8
a) the first term is what cubed? ___ x ___
b) the last term is what cubed? ___ 2 ___
c) the second term is divisible by __ 3 ____ and __ 2 ____.
d) the third term is divisible by ___ 3 ____ and ____ 2 ___2.
So, x3 + 6x2 + 12x + 8 can be factored as (x + 2)3.
Ex2: x3 + 12x2 + 48x + 64
a) the first term is what cubed? __ x ____
b) the last term is what cubed? ___ 4 ___
c) the second term is divisible by __ 3 ____ and __ 4 ____.
d) the third term is divisible by ___ 3 ____ and ___ 4 ____2.
So, x3 + 12x2 + 48x + 64 can be factored as (x + 4)3.
Ex3: x3 + 30x2 + 100x + 1000
a) the first term is what cubed? __ x ____
b) the last term is what cubed? __ 10 ____
c) the second term is divisible by __ 3 ____ and __ 10 ____.
d) the third term is divisible by __ 3 _____ and ___ 10 ____2. NO!
So, x3 + 30x2 + 100x + 1000 is __ PRIME ___________.
**The middle two terms MUST be divisible by 3 in order for the polynomial to factor.**
Now you try…
1. x3 + 15x2 + 75x + 125
a) the first term is what cubed? __ x ____
b) the last term is what cubed? __ 5 ____
c) the second term is divisible by __ 3 ____ and __ 5 ____.
d) the third term is divisible by __ 3 _____ and __ 5 _____2.
______ (x + 5)3 _______________________
2. n3 + 12n2 + 24n + 64
a) the first term is what cubed? __ n ____
b) the last term is what cubed? __ 4 ____
c) the second term is divisible by __ 3 ____ and ___ 4 ___.
d) the third term is divisible by ___ 3 ____ and ___ 4 ____2. NO!
________ PRIME _____________________
3. n3 + 18n2 + 108n + 216
a) the first term is what cubed? __ n ____
b) the last term is what cubed? __ 6 ____
c) the second term is divisible by __ 3 ____ and ___ 6 ___.
d) the third term is divisible by ___ 3 ____ and ___ 6 ____2.
________ (x + 6)3 _____________________
Binomials
(a – b)3
Polynomial
(Combine Like Terms)
a3 – 2a2b + 2ab2 – a2b + ab2 – b3
a3 – 3a2b + 3ab2 – b3
x3 – 2x2 + 1x – 1x2 + 2x – 1
x3 – 3x2 + 3x – 1
(a – b)∙(a – b)∙(a – b)
(a –
(x – 1)3
Distribute Several Times
b)(a2
– 2ab +
b2)
(x – 1)∙(x – 1)∙(x – 1)
(x – 1)(x2 – 2x + 1)
What do you notice between the polynomial generated from (a + b)3 and (a – b)3?
The 2nd and 4th terms are negative in the (a – b)3 polynomial.
Always check when there are negatives to see if the 2nd and 4th terms are negative. If there
are any discrepancies then this is not a cubic that will factor.
Do the following polynomials fit the rule for factoring by a binomial cubed?
Ex1: x3 – 9x2 + 27x – 27
a) the first term is what cubed? __ x ____
b) the last term is what cubed? __ 3 ____
c) the second term is divisible by __ 3 ____ and __ 3 ____.
d) the third term is divisible by __ 3 _____ and ___ 2 ____2.
So, x3 – 9x2 + 27x – 27 can be factored as (x – 3)3.
Ex2: x3 – 15x2 + 75x – 125
a) the first term is what cubed? _ x __
b) the last term is what cubed? __ 5 ____
c) the second term is divisible by __ 3 ____ and __ 5 ____.
d) the third term is divisible by ___ 3 ____ and ___ 5 ____2.
So, x3 – 15x2 + 75x – 125 can be factored as (x – 5)3.
Try on your own…
4.) x3 – 3x2 + 3x – 1
a) the first term is what cubed? __ x ____
b) the last term is what cubed? __ 1 ____
c) the second term is divisible by __ 3 ____ and __ 1 ____.
d) the third term is divisible by ___ 3 ____ and __ 1 _____2.
______ (x – 1)3 _______________________
5.) x3 – 6x2 + 12x – 8
a) the first term is what cubed? __ x ____
b) the last term is what cubed? __ 2 ____
c) the second term is divisible by __ 3 ____ and __ 2 ____.
d) the third term is divisible by ___ 3 ____ and ___ 2 ____2.
_______ (x – 2)3 _____________________
6.) x3 – 12x2 – 48x – 64
a) the first term is what cubed? __ x ____
b) the last term is what cubed? __ 4 ____
c) the second term is divisible by __ 3 ____ and __ 4 ____.
d) the third term is divisible by ___ 3 ____ and ___ 4 ____2.
______ PRIME (-48x)_______________________