1. MTH-4106 Pretest Z -54 = (x – 9y)(x + 6y) -3 = 18x2 + 12x – 33x – 22
For questions 1 to 7 express the polynomials using their most basic factors.
-54
1. x2  3xy  54y2 +1
-54
= (x – 9y)(x + 6y) +2
-27 -3 +3
-18 +6 -9
x2  21x  22
(18)(-22)= -396
= 18x2 + 12x – 33x – 22
= 6x(3x + 2) – 11(3x + 2)
= (3x + 2)(6x – 11)
-396 +1
-21
-198 +2
-132 +3
-99 +4
-66 +6
-44 +9
-36
+11
-33
+12
-22
+18
a2 – 12a + 45ab  9b
= 3(20a2 – 4a + 15ab – 3b)
= 3(4a(5a – 1) + 3b(5a – 1))
= 3(5a – 1)(4a + 3b)

2. = a2(x2 + 2xy + y2) – x2(x2 + 2xy + y2) = (x2 + 2xy + y2)(a2 – x2)
a4c2 - 63b6c2
= 7c2(4a4 – 9b6)
= 7c2(2a2 + 3b3)(2a2 – 3b3)
a2xy - x2y2 + a2y2 - x4 - 2x3y + a2x2
= a2x2 + 2a2xy + a2y2 – x4 – 2x3y – x2y2
= a2(x2 + 2xy + y2) – x2(x2 + 2xy + y2)
= (x2 + 2xy + y2)(a2 – x2)
= (x + y)(x + y)(a + x)(a – x)
= (x + y)2(a + x)(a – x)
7. 9abx – 12abz - 18cx + 42cz – 9abz
= 9abx – 12abz – 18cx + 42cz – 9abz
= 9abx – 21abz – 18cx + 42cz
= 3(3abx – 7abz – 6cx + 14cz)
= 3(ab(3x – 7z) – 2c(3x – 7z))
= 3(3x – 7z)(ab – 2c)

3. 9x3 – x = x(9x2 – 1) = x(3x + 1)(3x + 1) 3x2 + 8x – 3
8. Reduce the following fraction to its lowest terms.
9x3 – x = x(9x2 – 1)
= x(3x + 1)(3x + 1)
3x2 + 8x – 3
= 3x2 – 1x + 9x – 3
= x(3x – 1) + 3(3x – 1)
= (3x – 1)(x + 3)
9. Determine the following product. Express answer in its simplest terms.
6x2 – x – 2
= 6x2 – 4x + 3x – 2
= 2x(3x – 2) + 1(3x – 2)
= (3x – 2)(2x + 1)
5x2 – 7x – 6
= 5x2 – 10x + 3x – 6
= 5x(x – 2) + 3(x – 2)
= (x – 2)(5x + 3)

4. 10. Determine the following quotient
10. Determine the following quotient. Express answer in its simplest terms.
11. Determine the following sum. Express answer in its simplest terms.

5. 12. Determine the following difference
12. Determine the following difference. Express answer in its simplest terms.
13. Prove the following by working on one side.

6. 14. Prove the following by working on both sides.