Math 0361
Test 5 Review
Definitions:
1. Factored Form – An expression is written as a product of one or more terms. (Lial G-3)
Example: 2x+2 = 2(x+1)
2. Common Factors(Divisors) – An integer that is a factor of two or more integers. (Lial G-1)
Example: 4 and 6 have a common factor of 2.
3. Greatest Common Factor(Greatest Common Divisor) – The largest factor of all the terms of a
polynomial or list of integers.(Lial G-3)
Example: 12x and 30x2 have a greatest common factor of 6x.
4. Prime Polynomials – A polynomial that cannot be factored into factors having only integer
coefficients.(Lial G-5)
5. FOIL Method – A method for multiplying two binomials (a+b)(c+d). (Lial G-3)
Example: (a+b)(c+d)= ac +ad +bc +bd
6. Difference of Squares – The difference of two squared terms. (Lial G-2)
Example: x2 – y2 = (x+y)(x-y)
7. Difference of Cubes – The difference of two cubed terms.(Lial G-2)
Example: x3 – y3 = (x-y)(x2+xy+y2)
8. Sum of Cubes - The sum of two cubed terms. (Lial G-7)
Example: x3+y3=(x+y)((x2-xy+y2)
9. Perfect Square Trinomial – A trinomial that can be factored as the square of a binomial.(Lial G-5)
10. Standard Form of a Quadratic Equation – an equation that can be written in the form ax2 + bx + c
= 0, where a,b, and c are real numbers, with a  0. (Lial G-7)
Example: 2x2 + 4x + 8 = 0
11. Zero-Factor Property – If two numbers have a product of 0, then at least one of the numbers is
0. (Lial G-8)
Helpful Charts:
Divisibility Chart (Lial 296):
Finding the Greatest Common Factor (Lial 296):
Step 1: Factor. Write each number in prime factored form.
Step 2: List common factors. List each prime number or each variable that is a factor of every term in
the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest
common factor.)
Step 3: Choose least exponents. Use as exponents on the common prime factors the least exponent
from the prime factored forms.
Step 4: Multiply. Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest
common factor is 1.
Factor a Polynomial with Four Terms by Grouping (Lial 300):
Step 1: Group terms. Collect the terms into two groups so that each group has a common factor.
Step 2: Factor within groups. Factor out the greatest common factor from each group.
Step 3: Factor the entire polynomial. Factor out a common binomial factor from the results of Step 2.
Step 4: If necessary, rearrange terms. If Step 2 does not result in a common binomial factor,
try a different grouping.
Guidelines for Factoring ax2 + bx + c (Lial 306):
1. Both integers must be positive if b and c are positive.
2. Both integers must be negative if c is positive and b is negative.
3. One integer must be positive and one must be negative if c is negative.
FOIL Method:
Special Factorizations (Lial 323):
Solving a Quadratic Equation by Factoring (Lial 331):
Step 1: Write the equation in standard form — that is, with all terms on one side of the equals sign
in descending power of the variable and 0 on the other side.
Step 2: Factor completely.
Step 3: Use the zero-factor property to set each factor with variable equal to 0, and solve the
resulting equations.
Step 4: Check each solution in the original equation.
Problems:
1. Complete the factoring by writing the given polynomial as the product of two factors.
-14x8 =-7x3 ( ___ )
2. Complete the factoring.
20a2b + 25a3b6 = 5a2b(______________)
3. Factor.
8x3 – 6x2 + 2x
4. Write the following in factored form by factoring out the greatest common factor.
y2(x-6) + 1(x-6)
5. Is the expression in factored form or not? If it is not in factored form, factor it if possible.
17x2(y+5) + 9(y-5)
6. Factor by grouping.
Y2+15x+3y+5xy
7. Factor by grouping.
7z3+7zb2+8z2b+8b3
8. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
x2-9x+18
9. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
x2+5x-14
10. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
x2+x+30
11. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
-8+7x+x2
12. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
b2+5bf+6f2
13. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
x2+13xy+40y2
14. Factor the given polynomial completely. If the polynomial cannot be factored, say that it is
prime.
2x3+10x2+8x
15. Find the special product.
2 

 x   x 
3 

2

3
16. Factor by grouping.
9m3-3m2p2-3mp+p3
17. Factor completely.
a2-49
18. Factor the following binomial completely. If the binomial is prime, say so.
3n2+9
19. Factor completely.
36x2-25y2
20. Factor.
100s2+160sp+64p2
21. Factor.
2
3
t2+ t 
1
9
22. Factor completely.
27u3+8g4
23. Factor the trinomial completely.
9x2+2x+
1
9
24. Write the following in factored form by factoring out the greatest common factor.
m(m+8n)+9n(m+8n)
25. Solve using the zero-factor property.
6s(11s+5)=0
26. Solve.
w2=-18-9w
27. Solve the equation, and check your solution.
9s2+42s=-49
28. Solve the equation.
(3x+4)(3x2-13x+4)=0
29. Solve the equation.
y3-13y2+40y=0
30. Solve the equation.
(2x-3)(x+6)=(x-6)(x+3)
Works Cited
Lial, Hornsby, and McGinnis. Beginning and Intermediate Algebra, 5th ed. Boston: Pearson, 2012. Print.