Review of basic math and college algebra
multiplication tables
order of operations – the rules that tell us the sequence in which we should solve an arithmetic expression with
multiple operations.
11 − 5 + 3 ÷ 3 · 42 · (2 − 1) =
1 + 2 · �4 − 3 · (1 + 23 )� =
answer: 22
answer: −45
1
associative property of addition – the result is the same regardless of the order you add three or more numbers.
2+3+4= 3+4+2
=4+2+3
=3+2+4
relationship between addition and subtraction example
7 − 5 = 7 + (−5)
= −5 + 7
associative property of multiplication – the result is the same regardless of the order you multiply three or more
numbers.
2·3·4 =3·4·2
= 4·2·3
= 3·2·4
relationship between multiplication and division example. fraction bars are like division symbols.
5÷2 =
So,
5 5 1
1
= · =5·
2 1 2
2
5÷2= 5·
1
2
distributive property examples
3 ∙ (2 + 5) = 3 ∙ 2 + 3 ∙ 5
7(𝑥𝑥 + 𝑦𝑦 + 3) = 7𝑥𝑥 + 7𝑦𝑦 + 21
factor – a number or algebraic expression that divides another number or expression evenly, i.e., with no remainder.
4 is a factor of 120
𝑥𝑥 + 2 is a factor of 𝑥𝑥 2 + 2𝑥𝑥
integer factorization – expressing a positive integer as the product of smaller integers.
4 · 30 = 120
Therefore, 4 · 30 is an integer factorization of 120.
prime numbers – positive integers that are divisible only by 1 and the number itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are prime numbers.
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prime factorization – factorization of a positive integer with prime numbers only.
The prime factorization of 600 is
2 · 2 · 2 · 3 · 5 · 5 = 2 3 · 3 · 52
fractions
Multiply the two numerators and the two denominators when multiplying two fractions.
2 4 2·4
∙ =
3 7 3·7
=
8
21
Cancel common factors in the numerator and denominator when reducing a fraction to lowest terms.
25 5 · 5
=
40 5 · 8
=
5 5
·
5 8
= 1·
=
5
8
5
8
Rewrite two fractions with a common denominator when adding or subtracting two fractions.
5 3 5·2 3·3
+ =
+
3 2 3·2 2·3
=
=
=
10 9
+
6 6
10 + 9
6
19
6
3
Rewrite as the product of two fractions when dividing two fractions. Fraction bars are like division symbols.
2
5 = 2÷3
3
5 7
7
=
=
keep the top fraction. switch division to multiplication. flip the bottom fraction.
2 7
·
5 3
14
15
factoring an algebraic expression with greatest common factor (GCF)
factor the algebraic expression 20𝑥𝑥 4 − 15𝑥𝑥 3 + 30𝑥𝑥 2
the expressions 20𝑥𝑥 4 , −15𝑥𝑥 3 , 30𝑥𝑥 2 are called terms
the numbers 20, −15, and 30 are called coefficients
the greatest common factor of the coefficients is 5. factor it out. then by the distributive property
20𝑥𝑥 4 − 15𝑥𝑥 3 + 30𝑥𝑥 2 = 5(4𝑥𝑥 4 − 3𝑥𝑥 3 + 6𝑥𝑥 2 )
find the greatest common factor of the variable 𝑥𝑥 by finding the lowest power of 𝑥𝑥 that will divide all terms and
factor it out. the greatest common factor of the variable 𝑥𝑥 is 𝑥𝑥 2 . factor it out. then by the distributive property and
the rules of exponents.
20𝑥𝑥 4 − 15𝑥𝑥 3 + 30𝑥𝑥 2 = 5𝑥𝑥 2 (4𝑥𝑥 2 − 3𝑥𝑥 + 6)
FOILing – expanding the product of two factors with each factor containing two unlike terms.
First Outer Inner Last
(𝑥𝑥 + 𝑑𝑑)(𝑥𝑥 + 𝑒𝑒) = 𝑥𝑥 2 + 𝑒𝑒𝑒𝑒 + 𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑
= 𝑥𝑥 2 + (𝑒𝑒 + 𝑑𝑑)𝑥𝑥 + 𝑑𝑑𝑑𝑑
𝑑𝑑 and 𝑒𝑒 are integers
factoring trinomials of the form 𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒃𝒃 + 𝒄𝒄
𝑏𝑏 and 𝑐𝑐 are integers
if 𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 can be factored with integers, then
𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = (𝑥𝑥 + 𝑑𝑑)(𝑥𝑥 + 𝑒𝑒)
for some integers 𝑑𝑑 and 𝑒𝑒.
= 𝑥𝑥 2 + (𝑒𝑒 + 𝑑𝑑)𝑥𝑥 + 𝑑𝑑𝑑𝑑
So, 𝑏𝑏 = 𝑒𝑒 + 𝑑𝑑.
𝑐𝑐 = 𝑑𝑑𝑑𝑑.
So, ask yourself what two integers add to 𝑏𝑏 and multiply to 𝑐𝑐 when factoring trinomials of this form.
Practice – Factor the following trinomials.
𝑥𝑥 2 + 8𝑥𝑥 + 12
𝑥𝑥 2 − 7𝑥𝑥 − 18
𝑥𝑥 2 − 9𝑥𝑥 + 14
𝑥𝑥 2 − 10𝑥𝑥 + 25
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factoring trinomials of the form 𝒂𝒂𝒂𝒂𝟐𝟐 + 𝒃𝒃𝒃𝒃 + 𝒄𝒄
𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 are integers.
There are different methods for factoring a trinomial of this form if it can be factored with integers.
Three known methods are the box method, the X method, and the Slide Divide Slide method.
Each method requires the product 𝑎𝑎𝑎𝑎.
Factor the trinomial 𝟕𝟕𝟕𝟕𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟖𝟖
𝑎𝑎 = 7
𝑏𝑏 = −26
𝑐𝑐 = −8
in this example
One way of finding the factors for this trinomial is by trial and error. The product of the first terms has to equal
𝟕𝟕𝟕𝟕𝟐𝟐 and the product of the last terms has to equal −𝟖𝟖. The sum of the inner and outer terms has to equal −𝟐𝟐𝟐𝟐𝟐𝟐.
Slide Divide Slide method for finding the factors.
Slide the 𝑎𝑎 to the end of the trinomial, multiply it by 𝑐𝑐, then factor
𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒃𝒃 + 𝒂𝒂𝒂𝒂 = 𝒙𝒙𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟖𝟖 · 𝟕𝟕
= 𝒙𝒙𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟓𝟓𝟓𝟓
= (𝒙𝒙 − 𝟐𝟐𝟐𝟐)(𝒙𝒙 + 𝟐𝟐)
Divide the last term (integer) in each factor by the value of 𝑎𝑎 and reduce the fraction to lowest terms.
�𝒙𝒙 −
𝟐𝟐𝟐𝟐
𝟐𝟐
𝟐𝟐
� �𝒙𝒙 + � = (𝒙𝒙 − 𝟒𝟒) �𝒙𝒙 + �
𝟕𝟕
𝟕𝟕
𝟕𝟕
Slide the denominator in a fraction to the front of the 𝑥𝑥 in the factor. Now you have the factors for the trinomial
𝟕𝟕𝟕𝟕𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟖𝟖 = (𝒙𝒙 − 𝟒𝟒)(𝟕𝟕𝟕𝟕 + 𝟐𝟐)
FOIL your answer to verify you have the correct factors.
Practice – Factor the following trinomials. Answers are on the next page.
3𝑥𝑥 2 − 17𝑥𝑥 + 20
9𝑥𝑥 2 + 15𝑥𝑥 + 4
relationship between percentages, decimals, and fractions example
21.47% = 0.2147 =
2147
10000
5
answers for practice problems from previous page
3𝑥𝑥 2 − 17𝑥𝑥 + 20 = (3𝑥𝑥 − 5)(𝑥𝑥 − 4)
9𝑥𝑥 2 + 15𝑥𝑥 + 4 = (3𝑥𝑥 + 1)(3𝑥𝑥 + 4)
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