Multiplying and Factoring with Algebra Tiles
The Basics Building Blocks click here to see adding examples
1*1=1 1*x=x x*x=x2
1 1 x
An Example of Multiplying
(x+2)(x+3) = x2+5x+6
Answer: FOIL: First= x2 Outside + Inside = 3x+2x = 5x Last=2*3=6
Note: the width is x+2.
The
height is x+3.
x + 2
The area is x2+5x+6
x
+
The Same Example, with Factoring
To see that x2+5x+6 = (x+2)(x+3)
Start with x2 +5x +6
And rearrange the blocks until you can make a rectangle, the same one you see above
. x + 2
x
+
Factoring method from the book:
For ax2+bx+c = x2+5x+6, find a pair of factors of ac=1*6=6 that sum to b=5:
Factors of 6 Sum to 5
2,3 5 Notice there are 2 vertical rectangles and 3 horizontal rectangles, for a total of 5x.
2x+3x=5x and 2*3=6.
More examples
(Note: Start students with all positive coefficients. Negative coefficients are possible, but might be better explained after they understand the positive ones. See examples with negatives-- in two colors-- on the other sites).
Example #1
(2x+1)(x+3) = 2x2+7x+3
Answer: FOIL: First= 2x2 Outside + Inside = 6x+1x = 7x Last=1*3=3
2x + 1
x
+
3
For ax2+bx+c = 2x2+7x+3, find a pair of factors of ac=2*3=6 that sum to b=7:
Factors of 2*3=6 Sum to 7
6, 1 7 Notice there are 6 horizontal rectangles and 1 vertical rectangle, for a total of 7x.
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Example #2
(2x+3)(3x+2)=6x2+13x+6
Answer: FOIL: First= 2x*3x= 6x2 Outside + Inside=4x+9x=13x Last=3*2=6
2x + 3
3x
+
2
For ax2+bx+c = 6x2+13x+6, find a pair of factors of ac=6*6=36 that sum to b=13:
Factors of 6*6=36 Sum to 13
9, 4 13 Notice there are 9 vertical rectangles and 4 horizontal rectangles, for a total of 13x.
Now try some examples yourself
Question: Factor 3x2+11x+6 Try both yourself and click here for both answers
Question: Factor 6x2+17x+5