Algebra Review
Warm-up (3 m)
• Multiply:
1. 4x2(7x3 - 6x2 + 12x - 10)
• Factor:
3. x3 – 64x
2. (3x2 - 5)(x + 4)
4. 9x2 – 9x – 4
We’re going to review the following
skills for the next unit:
•
•
•
•
•
•
Multiplying Polynomials
Factoring Polynomials
Simplifying Rational Expressions
Multiplying Rational Expressions
Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Multiplying Polynomials
Distribute
• Multiply each term inside
the parentheses by the
term outside the
parentheses.
3x5(x7 – 2x4 + 11x)
FOIL
• First – Outer – Inner – Last
(3x + 5)(x – 7)
Multiplying Polynomials, cont.
Vertical Multiplication
• Works well when you multiply anything larger
than a binomial and a binomial.
– Example: (3x2 – x + 1)(x2 + 2x – 3)
• Very similar to long multiplication by hand.
Example
• (7x2 – 5x +6)(2x – 1)
Example
• (3x2 – x + 1)(x2 + 2x – 3)
Multiplying with Trigonometric
Functions
• Exactly the same as multiplying without
trigonometric functions.
tanx(tan x  2tanx  3)
2
Your Turn:
• Multiply problems 1 – 10 in the Algebra
Review packet
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Factoring
• Remember, there are four types of factoring
that we reviewed at the beginning of the
semester:
– Leading Coefficient = 1 (“Regular” Factoring)
– Leading Coefficient ≠ 1 (Box Method or Welsh
Method)
– Greatest Common Factor (GCF)
– Difference of Squares
Leading Coefficient = 1
x2 – 7x + 10
Leading Coefficient ≠ 1
3x2 – 11x – 20
Greatest Common Factor
4x4 – 40x3 + 8x2
Difference of Squares
81x4 – 100
Factoring with Trigonometric
Functions
• Exactly the same as factoring without
trigonometric functions.
s i nx cosx  2cos x
2
sin x 4
2
Your Turn:
• Factor problems 11 – 24 in the Algebra Review
packet.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Warm-up (3 m)
1. Multiply:
2. Factor: cot x  sec x
4
(tan2 x  4 tanx  2)(2 tanx  5)
• Find the reciprocals of the numbers below:
3.
6x
11y
4. 7
4
Seek and Solve!!!
Show all your work on a piece of
paper. I’m collecting it for a
classwork grade.
Simplifying Rational Expressions
• You can only cancel factors that are separated
by multiplication!!!
Wrong!!!
x5 5

x 7 7
Right!!!
(x  2)(x  9) x  2

(x  7)(x  9) x  7
Simplifying Rationals, cont.
• You can also reduce factors – as long as
they’re separated by multiplication.
5x
2
10x
2
4
6g m
7 2
4g m
Simplifying Rationals, cont.
1. Factor the numerator and the denominator
2. Optional – Identify the factors in the
numerator and the denominator.
3. Cancel common factors in the numerator and
the denominator.
Example
2x  4x  48x
2
4x  22x  12
3
2
Factors in Numerator
Factors in Denominator
Example
cot x  1
s i nx cotx  s i nx
2
Factors in Numerator
Factors in Denominator
Your Turn:
• Simplify problems 25 – 32 in the Algebra
Review packet. Remember to factor the
numerator and the denominator first, AND
you can only cancel factors separated by
multiplication.
25.
26.
27.
28.
29.
30.
31.
32.
Warm-up (3 m)
6 cos x  6 s i n x
1. Simplify:
2x cos2 x  2x s i n2 x
4
4
6 cos x  6 s i n x
2x cos2 x  2x s i n2 x
4
4
Homework Review
Multiplying Rational Expressions
1. Factor the numerator and the denominator.
2. Cancel and/or cross cancel any common
factors that are separated by multiplication.
3. Optional – Rewrite the simplified fractions.
4. Multiply across. (Multiply the numerators
together and the denominators together.)
Example
8x  16
x

5x
5x  10
3
Example
cos2 x  10cos 25 cos2 x  3cosx

2
cosx  5
cos x  9
Your Turn
• Multiply problems 33 – 38 in the Algebra
Review packet. Simplify your answers.
33.
34.
35.
36.
37.
38.
Warm-up (4 m)
2 sinx  10
4 sinx  24
 2
1. Multiply:
2
sin x  8 sinx  12 sin x  25
2 sinx  10
4 sinx  24
 2
2
sin x  8 sinx  12 sin x  25
What About…?
x  4x x  x

4
3
x x
4x  x
3
4
Dividing Rational Expressions
• Division is the same thing as multiplication by
the reciprocal!
10
5
1
1
2
3
Dividing Rationals, cont.
1. Rewrite the division as multiplication by the
reciprocal.
2. Factor the numerator and the denominator.
3. Cancel and/or cross cancel any common
factors separated by multiplication.
4. Multiply across.
Example
5
9x
2
8y
3x
9
16y
5x  10
8
x
x 2
3
x
Example
Your Turn:
• Divide problems 39 – 48 in the Algebra Review
packet.
39.
40.
41.
42.
43.
44.
45.
46.
Adding and Subtracting Rational
Expressions
• If the fractions have the same denominator,
add or subtract the numerators. (Make sure
to distribute the subtraction sign!!!)
• Simplify the fraction is possible.
Examples
x  5 x  6x  21

x 7
x 7
2
sinx  4 2 sinx  5

sinx  2 sinx  2
Adding and Subtracting Rational
Expressions, cont.
• If the fractions have the different denominators,
1. Factor the numerator and the denominator.
2. Simplify each fraction individually if possible.
3. Compare the denominators of each fraction. Identify
the “missing” factors from each fraction. (Finding the
Least Common Denominator)
"mi ss i ng" fa ctors
4. Multiply each fraction by 1. Rewrite 1 as
"mi ss i ng" fa ctors
Adding and Subtracting Rational
Expressions, cont.
5.
6.
7.
8.
9.
Multiply across.
Combine all fractions into one fraction.
Simplify the numerator.
Factor the numerator if possible.
Simplify/reduce the fraction if possible.
Examples in Smart Board File
Example
x 5 x 7

x 2 x  4
Example
tanx  3
tanx  1

tanx  10 tanx  15
Example
s i n2 x  s i nx  12
4 s i nx  12

2
s i n x  2s i nx  8 2s i n2 x  2s i nx  4
Your Turn:
• Add or subtract problems 49 – 62 in the
Algebra Review packet.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
Activity!!!