(1)Factor.
(a) 25x2 - (40/3)x + 16/9 (b) z3 - z2 - 13z + 4 (c) (x + 5)2 - 64 (d) 102B -(9)(10)B + 14 (e) 3x3 + 4x2 - 5x - 2SOLUTIONS:
(a) 25x2 - (40/3)x + 16/9
Constant term, 16/9 is a perfect square: sqrt(16/9) = 4/3
25x2 is also a perfect square: sqrt(25x2) = 5x
2(4/3)(5x) = (40/3)x [refer to hint] which equals the middle term
Therefore, it is a perfect square and can be factored into (5x - 4/3)2.
Check: (5x + 4/3)2 = (5x - 4/3)(5x - 4/3)
= 25x2 + (20/3)x + (20/3)x + 16/9
= 25x2 - (40/3)x + 16/9
It works!
(b) z3 - z2 - 13z + 4
Factors of 4 are: 1, -1, 2, -2, 4, -4 (possible k values for (z - k)
as a factor)
Possible factors are: (z + 1), (z - 1)
(z + 2), (z - 2)
(z + 4), (z - 4)
Checking by substituting according to the rules outlined:
(z + 1): k = -1, sub in x = -1 into ploynomial:
(-1)3 - (-1)2 -13(-1) + 4 = 15
(z - 1): k = 1, sub in x = 1:
(1)3 - (1)2 -13(1) + 4 = -9
(z + 2): k = -2, sub in x = -2:
(-2)3 - (-2)2 -13(-2) + 4 = 18
(z - 2): k = 2, sub in x = 2
(2)3 - (2)2 -13(2) + 4 = -26
(z + 4): k = -4, sub in x = -4:
(-4)3 - (-4)2 -13(-4) + 4 = -24
(z - 4): k = 4, sub in x = 4:
(4)3 - (4)2 -13(4) + 4 = 0
Thus, the only one of these that is correct is (z - 4). We must then use
long division to find the other factor:
z2 + 3z - 1
-------------------
(z - 4) | z3 - z2 - 13z + 4
z3 - 4z2
---------
3z2 - 13z
3x2 - 12z
--------
-z + 4
-z + 4
-------
0
Therefore, z3 - z2 - 13z + 4 = (z - 4)(z2 + 3z - 1)
(c) (x + 5)2 - 64
This is of the form a2x2 - b2 where a = 1, x =(x + 5), b = sqrt(64) = 8 (in this case)
Thus, this expression is a "difference of squares" and can be factored into
(ax + b)(ax - b).
so, (x + 5)2 - 64 = [(x + 5) + 8][(x + 5) - 8]
= (x + 13)(x - 3)
(d) 102B -(9)(10)B + 14
let x = 10B, so the expression now becomes: x2 - 9x + 14
Use method of decomposition:
factors of 14 that add up to -9 are: -7 and -2
= x2 - 7x - 2x - 14
= x(x - 7) - 2(x - 7)
= (x - 7)(x - 2)
But since x = 10B, the original can be factored into (10B - 7)(10B - 2)
(e) 3x3 + 4x2 - 5x - 2
The fact that x = 1 is a root almost jumps out at you since
3(13) + 4(12) - 5(1) - 2 = 0
So x - 1 is a factor. Use long division to find the other factor:
3x2 + 7x + 2
--------------------
(x - 1)| 3x3 + 4x2 - 5x - 2
3x3 - 3x2
--------
7x2 - 5x - 2
7x2 - 7x
---------
2x - 2
2x - 2
--------
0
Therefore the other factor is 3x2 + 7x + 2, but it can be factored further into
(3x + 1)(x + 2) [use decomposition method]. Thus, the final answer is:
3x3 + 4x2 - 5x - 2 = (x - 1)(3x + 1)(x + 2)
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