Transcript 7-8 Special Products

Objective
The student will be able to:
use patterns to multiply special
binomials.
There are formulas (shortcuts) that
work for certain polynomial
multiplication problems.
2
b)
2
a
2
b
(a +
= + 2ab +
(a - b)2 = a2 – 2ab + b2
2
2
(a - b)(a + b) = a - b
Being able to use these formulas will help you in
the future when you have to factor. If you do not
remember the formulas, you can always multiply
using distributive, FOIL, or the box method.
Let’s try one!
1) Multiply: (x + 4)2
You can multiply this by rewriting this as
(x + 4)(x + 4)
OR
You can use the following rule as a shortcut:
(a + b)2 = a2 + 2ab + b2
For comparison, I’ll show you both ways.
1) Multiply (x + 4)(x + 4)
First terms: x2
Outer terms: +4x
Inner terms: +4x
Last terms: +16
Combine like terms.
x2 +8x + 16
Notice you
have two of
the same
answer?
x
x
+4
x2
+4x
+4 +4x +16
Now let’s do it with the shortcut!
1) Multiply: (x + 4)2
That’s why
the 2 is in
the formula!
using (a + b)2 = a2 + 2ab + b2
a is the first term, b is the second term
(x + 4)2
a = x and b = 4
Plug into the formula
a2 + 2ab + b2
(x)2 + 2(x)(4) + (4)2
This is the
Simplify.
same answer!
x2 + 8x+ 16
2) Multiply: (3x + 2y)2
using (a + b)2 = a2 + 2ab + b2
(3x + 2y)2
a = 3x and b = 2y
Plug into the formula
a2 + 2ab + b2
(3x)2 + 2(3x)(2y) + (2y)2
Simplify
9x2 + 12xy +4y2
Multiply (2a + 3)2
1.
2.
3.
4.
4a2 – 9
4a2 + 9
4a2 + 36a + 9
4a2 + 12a + 9
Multiply: (x – 5)2
using (a – b)2 = a2 – 2ab + b2
Everything is the same except the signs!
(x)2 – 2(x)(5) + (5)2
x2 – 10x + 25
4) Multiply: (4x – y)2
(4x)2 – 2(4x)(y) + (y)2
16x2 – 8xy + y2
Multiply (x – y)2
1.
2.
3.
4.
x2 + 2xy + y2
x2 – 2xy + y2
x2 + y2
x2 – y2
5) Multiply (x – 3)(x + 3)
First terms: x2
Outer terms: +3x
Inner terms: -3x
Last terms: -9
Combine like terms.
x2 – 9
Notice the
middle terms
eliminate
each other!
x
-3
x2
-3x
+3 +3x
-9
x
This is called the difference of squares.
5) Multiply (x – 3)(x + 3) using
2
2
(a – b)(a + b) = a – b
You can only use this rule when the binomials
are exactly the same except for the sign.
(x – 3)(x + 3)
a = x and b = 3
(x)2 – (3)2
x2 – 9
6) Multiply: (y – 2)(y + 2)
(y)2 – (2)2
y2 – 4
7) Multiply: (5a + 6b)(5a – 6b)
(5a)2 – (6b)2
25a2 – 36b2
Multiply (4m – 3n)(4m + 3n)
1.
2.
3.
4.
16m2 – 9n2
16m2 + 9n2
16m2 – 24mn - 9n2
16m2 + 24mn + 9n2