Warm-Up Factor the following trinomials. What do you notice? 1. x 2 - 6x + 9 2. 4x 2 - 4x + 1.
The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2...
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Transcript of The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2...
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
4x3+8x2+2x
GCF?– Yes…
• 2x
Factor it out…– 2x(2x2+4x+1)
Factor further?– All done.
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
x2-64
GCF?– No.
Difference of Squares?– Yes.
Find the square roots of 1st & last terms (x+8)(x-8) More?
– No; all done.
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
x2+18x+81
GCF?– No.
Difference of squares?– No.
Factor? What are the factors of the 1st & last terms?
x2+18x+81 (cont’d)
X; 1,81; 9,9; 4, 27;… (x _ _)(x _ _) +81 means?
– Both the same sign
+18x means?– Both must be +
(x+_)(x+_)
x2+18x+81 (cont’d)
Then what? (x+_)(x+_)… (x+9)(x+9)
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
x2-4x+4
GCF?– No.
Difference of squares?– No.
Factor. Factors of 1st & last terms x | 2,2; 1,4
x2-4x+4 (cont’d)
(x_ _)(x_ _) +4 means?
– Both factors have the same sign
-4x means?– Both factors must be -
(x-_)(x-_)… Try it out… (x-2)(x-2)
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
3x2+6x+3
GCF?– 3
3(x2+2x+1) Factor Further?
– Yes.
Factor x2+2x+1 (x _ _)(x _ _)
3x2+6x+3 (cont’d)
+1 means?– Both signs are the same
+2x means?– Both factors must be sums (+)
Find the factors… (x + 1)(x + 1) Final Answer: 3(x+1)(x+1)
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
2x2+13x-7
GCF?– No.
Difference of squares?– No.
Factor… Factors of 2x2 are 2x,x
2x2+13x-7 (cont’d)
(2x _ _)(x _ _) Factors of 7: 1,7 -7 means?
– One +, one -
+13x means?– Larger product must be +
Try it...
2x2+13x-7 (cont’d)
(2x+1)(x-7)– Outer Product: -14x– Inner Product: +1x
• Sum: -13x
(2x-1)(x+7)– Outer Product: +14x– Inner Product: -1x
• Sum: +13x
2x2+13x-7 (cont’d)
(2x+1)(x-7)– Outer Product: -14x– Inner Product: +1x
• Sum: -13x
*(2x-1)(x+7)* This is the one!– Outer Product: +14x– Inner Product: -1x
• Sum: +13x
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
4x3-36x
GCF?– 4x
4x(x2-9) Factor Further?
– Yes; is (x2-9) a difference of squares?• Yes
Find the square roots of the 1st & last terms
4x3-36x (cont’d)
Roots: x, 3 4x(x _ 3)(x _ 3) 4x(x+3)(x-3) All done!
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
3x2-7x+1
GCF?– No.
Difference of squares?– No.
Factor?– 3x,x | 1,1
Try the various combinations...
3x2-7x+1 (cont’d)
(3x+1)(x-1)– Products: -3x, +1x
• Sum: -2x
(3x-1)(x+1)– Products: +3x, -1x
• Sum: +2x
No matches, therefore 3x2-7x+1 is prime.
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10
3x2-11x+10
GCF?– No.
Difference of squares?– No.
Factors…– 3x, x | 1,10; 2,5
Try them...
3x2-11x+10 (cont’d)
(3x-1)(x-10)– Products: -30x, -x
• Sum: -31x
(3x-10)(x-1)– Products: -3x, -10x
• Sum: -13x
3x2-11x+10 (cont’d)
(3x-2)(x-5)– Products: -15x, -2x
• Sum: -17x
(3x-5)(x-2)– Products: -6x, -5x
• Sum: -11x
3x2-11x+10 (cont’d)
(3x-2)(x-5)– Products: -15x, -2x
• Sum: -17x
*(3x-5)(x-2)* This is the one!– Products: -6x, -5x
• Sum: -11x
It works!
The Factoring Puzzle
4x3+8x2+2x x2-64 x2+18x+81
x2-4x+4 3x2+6x+3 2x2+13x-7
4x3-36x 3x2-7x+1 3x2-11x+10