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Factor each polynomial.
1. 3x2 – 13x – 10
= (3x + 2)(x – 5)
Algebra 2
Review for
Ch #3 Test
2. 9a2 + 30ab + 25b2
= (3a + 5b)(3a + 5b)
a3 – b3 = (a – b)(a2 + ab + b2)
3. 4c2 – 81
= (2c + 9)(2c – 9)
4. 27n3 – 125
= (3n – 5)(9n2 + 15n + 25)
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Multiply each pair of polynomials.
5. (7y + 8)(7y – 8)
= 49y2 – 56y + 56y – 64 foil method
= 49y2 – 64 combine like terms
6. (4a + 5b)2
= (4a + 5b)(4a + 5b)
= 16a2 + 20ab + 20ab + 25b2 foil method
= 16a2 + 40ab + 25b2 combine like terms
7. (2y – 3)(4y2 + 6y + 9)
= 8y3 + 12y2 + 18y – 12y2 – 18y – 27
= 8y3 – 27 combine like terms
distributive property
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Factor each polynomial completely.
8. 98 – 32x2
= 2(49 – 16x2) factor 2 out
= 2(7 + 4x)(7 – 4x)
10. -24x2 – 6x + 45
= -3(8x2 + 2x – 15)
= -3(4x – 5)(2x + 3)
9. 2xy2 – 2xy – 12x
= 2x(y2 – y – 6) factor 2x out
= 2x(y – 3)(y + 2)
factor -3 out
11. 4a4 – a2 – 18
= (4a2 – 9)(a2 + 2) factor into 2 binomials
= (2a + 3)(2a – 3)(a2 + 2) {4a2 – 9 is a difference of two squares}
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Solve each equation.
12. a2 = 11a
13. 2x2 = 12 – 5x
-11a -11a
-12 + 5x -12 + 5x
a2 – 11a = 0
a(a – 11) = 0 factor a out
a = 0 or a – 11 = 0
a = 11
2x2 + 5x – 12 = 0
(2x – 3)(x + 4) = 0 factor into two binomials
2x – 3 = 0 or x + 4 = 0 set each equal to 0
+3
+3
2x = 3
x = 3/2
-4 -4
or
x=-4
14. 0 = a4 – 37a2 + 36
0 = (a2 – 36)(a2 – 1) factor into two binomials
0 = (a + 6)(a – 6)(a + 1)(a – 1) factor differences of two squares
a + 6 = 0 or a – 6 = 0 or a + 1 = 0 or a – 1 = 0 set each factor equal to 0
a = -6, 6, -1, 1
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15. One number is 6 less than another number. The product of the
smaller number and 3 more than the larger number is 10. Find
both numbers.
x = one number
x – 6 = other number
(x – 6)(x + 3) = 10
x2 – 3x – 18 = 10
foil method
-10 -10
x2 – 3x – 28 = 0
(x – 7)(x + 4) = 0 factor into two binomials
x – 7 = 0 or x + 4 = 0 set each factor equal to 0
+7
+7
x=7
x–6=1
-4
or
-4
x=-4
x – 6 = -10
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16. The length of the base of a triangle is 10cm less than 4 times the
height. Find the length of the base and the height of the triangle
if its area is 25 cm2.
25 = x(4x – 10)
2
25 = 4x2 – 10x
2
50 = 4x2 – 10x
-50
area of triangle = b • h
2
x
distributive property
multiply both sides by 2
-50
4x – 10
4x2 – 10x – 50 = 0
2(2x2 – 5x – 25) = 0 factor 2 out
2(2x + 5)(x – 5) = 0 factor into 2 binomials
2x + 5 = 0 or x – 5 = 0 set each factor equal to 0
-5
-5
+5
+5
2x = -5
x = 5 cm
x = - 5/2 4x – 10 = 10 cm
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17. Find three consecutive even integers such that the square of the
second integer, increased by twice the product of the first two
integers, is the same as 24 more than twice the square of the
first integer.
x = 1st
x + 2 = 2nd
x + 4 = 3rd
(x + 2)2 + 2x(x + 2) = 2x2 + 24
x2 + 4x + 4 + 2x2 + 4x = 2x2 + 24 distributive property
3x2 + 8x + 4 = 2x2 + 24 combine like terms
-2x2
-24
-2x2 -24
x2 + 8x – 20 = 0
(x + 10)(x – 2) = 0 factor into two binomials
x + 10 = 0 or x – 2 = 0 set each factor equal to 0
-10 -10
+2
+2
x = -10 or
x=2
x + 2 = -8
x+2=4
x + 4 = -6
x+4=6
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18. The length of a rectangle is 2m more than its width and 2m
shorter than the length of a diagonal. Find the length of the
diagonal and the area of the rectangle.
x+2
x+4
x
x2 + (x + 2)2 = (x + 4)2
x2 + x2 + 4x + 4 = x2 + 8x + 16 foil method
2x2 + 4x + 4 = x2 + 8x + 16 combine terms
-x2
-8x -16 -x2
-8x
-16
x2 – 4x – 12 = 0
(x + 2)(x – 6) = 0 factor into two binomials
x + 2 = 0 or x – 6 = 0 set each equal to 0
-2 -2
x = -2
+6
+6
x=6
x + 4 = 10m
Area = 48 m2
© Mr. Sims
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