Download Unit 8 ~ Contents

Unit 8 ~ Contents
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
8.16
Algebra Beauty and Awe ~ Boyle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Dividing a Trinomial by a Binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Factoring Completely. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Inverse Variation: y = kx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Adding and Subtracting Rationals With Common Denominators . . . . . . . . . . . . . . 12
Quiz 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Responding to Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Multiple Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Graphing Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Dividing Polynomials With Remainders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Lowest Common Multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Quiz 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
How’d He Do That? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Quadratic Equations: Solving by Factoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Solving Applications by Using Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . 41
Writing Equivalent Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Review for Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Unavoidable Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.2
Factoring Completely
To completely factor a polynomial may require more than one step. Sometimes the initial answer
can be factored again. The polynomial 18x 2 – 32 is an example.
Step 1, factor out the GCF (2).
2(9x 2 – 16)
Step 2, the binomial factor is the difference of two squares, so continue factoring.
(9x 2 – 16) = (3x + 4)(3x – 4)
The polynomial completely factored.
2(3x + 4)(3x – 4)
To Completely Factor Polynomials
1. Factor any GCF from the expression.
2. Look for the following special patterns and factor accordingly:
a. A difference of two squares.
b. A perfect square trinomial.
3. Factor any other trinomials, if possible.
4. Look for the possibility of factoring by grouping.
5. After each step, check the answer to see if it can be broken down any further.
Example 1. Factor the polynomial 3x 2y – 12y.
3x 2y – 12y
3y (x 2 – 4)
3y (x – 2)(x + 2)
Original polynomial.
GCF factored out, leaving a difference of squares.
Difference of squares factored.
Example 2. Factor the polynomial 12x 3+ 36x 2 + 27x.
12x 3+ 36x 2 + 27x
Original polynomial.
3x (2x + 3) (2x + 3)
Perfect square trinomial factored.
3x (4x 2 + 12x + 9)
GCF factored out, leaving a perfect square trinomial.
8.2 Factoring Completely ~ 5
Example 3. Factor the polynomial 8ax + 20ay + 8bx + 20by.
8ax + 20ay + 8bx + 20by
Original polynomial.
4 [ 2ax + 5ay + 2bx + 5by ]
GCF factored out.
4 [ a(2x + 5y) + b(2x + 5y) ]
GCF’s removed from binomials.
4[ (2ax + 5ay) + (2bx + 5by) ]
Polynomial grouped into two binomials.
4 [ (2x + 5y) (a + b) ]
Common binomial factored out.
4 (2x + 5y) (a + b)
Unnecessary brackets removed.
Example 4. Factor the polynomial x 4 – 18x 2 + 81.
x 4 – 18x 2 + 81
Original polynomial—a perfect square trinomial.
(x 2 – 9)(x 2 – 9)
Trinomial factored, leaving differences of squares.
(x – 3)2 (x + 3)2
Answer simplified with exponents.
(x – 3)(x + 3)(x – 3)(x + 3)
Differences of squares factored.
Example 5. Factor the polynomial x 16 – 1.
x 16 – 1
(x 8 + 1)(x 8 – 1)
Original polynomial—a difference of squares.
Binomial factored, leaving another difference of squares.
(x 8 + 1)(x 4 + 1)(x 4 – 1)
Binomial factored, leaving another difference of squares.
(x 8 + 1)(x 4 + 1)(x 2 + 1)(x 2 – 1
(x 8 + 1)(x 4 + 1)(x 2 + 1)(x + 1)(x – 1)
Binomial factored, leaving another difference of squares.
Final difference of squares factored.
Factor polynomials completely. 8.2
1. 5x 2 – 20x – 105
2. 6axy – 3bxy
3. 12x 2 – 46x + 14
7. 2x 5 – 10x 3 + 8x
8. 32x 4 – 40x 3 – 12x 2
9. 6x 2y – xy – y
4. 3x 4 – 48
6 ~ Algebra I Unit 8
5. 12x 2 + 96x + 192
6. x 8 – 1
Review
Divide. 8.1
10. (2x 2 – xy – y 2) ÷ (2x + y)
Factor by grouping. 7.3
12. 3rs + 6r – 4s – 8
11. (42x 2 + x – 1) ÷ (1 + 6x)
13. 8x 2 + 4x – 2xy – y
Divide the rational expressions. 7.7
15.
x 2 + 6x – 27
x 2 + 13x + 36
4x 2 – 9 2x + 3
÷
16.
2
x+2
x + 6x + 8
7x 3 ÷ 6x
14. 10xy + 25x + 14y + 35
17.
10x 3 5xy 2
3y ÷ 6
Solve for the requested information. 6.14
18. Chris’ jogging speed is 6 mph, whereas James’ speed is 5.5 mph. If they begin jogging at the same
time, how long would it be (in minutes) until their combined distance covered would be 4.6 miles?
Determine the excluded values for the expression. 6.9
3x
6
2
19. 4x 2 + 11x – 3
20. 3x + 2
21. 5x
5x + 2
22. 9x 2 – 4
Factor the trinomials. If any is not a perfect square, write not a perfect square. 6.3
23. 9w 2 – 12w + 4
24. 9 – 16x + 16x 2
25. m 2 + 2mn + n 2
Combine like radicals. If they cannot be combined, write cannot be combined. 7.9
26. 3 √π – 6 √π
27. 10 √x + 5 √x 2 – √x
Factor polynomials completely. 8.2
29. 5xy 3 – 30xy 2 + 40xy
30. 3x 5y – 48xy
3
28. 4 √7 + 5 √7
31. 8xy 3z + 12xyz
8.2 Factoring Completely ~ 7
8.3
Inverse Variation: y =
k
x
In some variations the dependent variable decreases as the independent variable increases. This
k
is called inverse variation and is represented by the general equation y = x . This equation is
expressed as, “y varies inversely as x.”
Inverse variation can be illustrated with a form of the distance equation: t = dr . The time it takes
to travel a specific distance varies inversely as the rate of travel. In other words, the faster you travel,
the less time it takes to reach your destination.
Inverse variation problems sometimes contain an unknown constant of proportionality. This
constant of proportionality can be found by substituting values for x and y in the inverse variation
k
equation: y = x .
Example 1. If y varies inversely as x, and y is 4 when x is 3, what is y when x is 2?
y=
4=
k
x
k
3
k = 12
General inverse variation formula.
Given values substituted for x and y.
Equation solved for k.
The constant of proportionality is 12, so the specific inverse equation is y =
y=
12
x
12
y= 2
y=6
When x is 2, y will be 6.
8 ~ Algebra I Unit 8
Specific equation.
2 substituted for x.
Equation solved for y.
12
.
x
Example 2. Time is inversely proportional as speed. If you normally bicycle to a neighboring city in
3.5 hours at 10 mph, how fast would you have to ride to get there in 2.5 hours?
t=
k
r
3.5 =
General inverse variation formula.
k
10
k = 35
Given values substituted for t and r.
Equation solved for k.
The constant of proportionality is 35, so the specific inverse equation is t =
t=
35
r
2.5 =
35
.
r
Specific equation.
35
r
r = 14
2.5 substituted for x.
Equation solved for r.
You would have to bicycle at 14 mph to reach the city in 2.5 hours.
Solve by using inverse variation. 8.3
1. If y varies inversely as x, and y is 5 when x is 3, what is y when x is 30?
2. If b varies inversely as a, and b is 12 when a is 7, what is b when a is 2?
3. If n varies inversely as m, and n is 25 when m is 2, what is n when m is 5?
4. If q varies inversely as p, and q is 6 when p is 6, what is q when p is 4?
5. Time is inversely proportional as speed. If Randy normally bicycles to his cousin’s house in 40
2
minutes ( hr) at an average of 12 mph, how fast would he have to ride to get there in 30 minutes?
3
6. If y varies inversely as x 2, and y is 2.5 when x is 2, what is y when x is 5?
7. For all rectangles of the same area the width varies inversely as the length. When the length for one
area is 15.5 feet the width is 8 feet. What is the width when the length is 20 feet?
8. The weight of an object varies inversely as the square of its distance from the center of the earth.
James weighs 200 pounds on the surface of the earth (at 4,000 miles from its center). How much
does he weigh in an airplane flying 5 miles high? Use a calculator and round to the nearest tenth.
8.3 Inverse Variation: y = x ~ 9
k
Review
Factor polynomials completely. 8.2
9. 4x 2y + 24x y + 36y
10. 2x 4 – 16x 2 + 32
11. 2x 2y – 18xy + 28y
Solve by using direct variation. 7.14
12. If y varies directly as x, and y is 3.75 when x is 4.5, what is y when x is 7?
13. The number of brownies for a certain recipe varies directly as the cups of flour used. If 2 cups of
flour yield 16 brownies, how many brownies would 5 cups of flour produce?
14. For snow, the rain equivalent in inches varies directly as the inches of snow that fell. If 13 inches of
average snow are equivalent to 1.3 inches of rain, what is the rain equivalent of 25 inches of snow?
Simplify the radicals. Show the steps. 6.8
15. √98x 7y 8
16. √338r 12s9
Divide. 8.1
18. (2e2 – 17e – 30) ÷ (2e + 3)
17. √192x 8
3
19. (12x 2 + 7x – 10) ÷ (4x + 5)
Graph the inequalities. Check each answer with a test point. 7.13
1
3
20. y < 5x – 2
21. y ≥ – x + 1
22. y ≥
5
x–3
2
Solve the systems of equations by the addition/elimination method. 6.12
23. 3x – 2y = 11
5x + 2y = 45
24. 3x – 5y = – 43
3x + 8y = 61
25. 6x – 2y = – 18
14x + 2y = – 22
Write equations for lines that have these slopes and points. 7.4
26. m = –
3
2
(–10, 3)
5
2
27. m = 8 ( , 4)
Multiply the rational expressions. 7.1
x+2
x+4
29. x 2 + 3x – 4 • x 2 – x – 6
10 ~ Algebra I Unit 8
30.
3–x
2x
x • x–3
28. m =
31.
6
5
(– 4, –7)
xy – 2x
6xy + 3y
• 10x + 5
x2
Solve the systems of equations by using the multiplication/addition method. 7.2
32. 2x – 3y = 5
3x + 2y = 14
33. 3x – 5y = –34
2x – 9y = – 51
34. 5x + 2y = 2
3x – 5y = –36
Solve by using inverse variation. 8.3
35. If y varies inversely as x, and y is 4 when x is 7, what is y when x is 2.5?
36. If b varies inversely as a, and b is 37.5 when a is 8, what is b when a is 25?
8.3 Inverse Variation: y = x ~ 11
k
8.4
Adding and Subtracting Rationals With
Common Denominators
To add or subtract rational expressions that share the same denominator, combine the
numerators and put the answer over their common denominator. After combining, check to see if
the numerator can be factored and cancel like factors.
Example 1.
3x
6x
+
5
5
=
3x + 6x
5
numerators combined over
common denominator
Example 2.
2x
2x 2
+
(x
+ 1)
(x + 1)
=
=
2x 2 + 2x
(x + 1)
numerators combined over
common denominator
9x
5
like terms
combined
=
2x (x + 1)
(x + 1)
numerator factored and
factors canceled
=
2x
simplified
answer
Example 3.
5x 2 + 13x
3x 2 + 7
–
2
(4x + 4x – 3)
(4x 2 + 4x – 3)
5x 2 + 13x – (3x 2 + 7)
(4x 2 + 4x – 3)
2x 2 + 13x – 7
(4x 2 + 4x – 3)
(2x – 1)(x + 7)
(2x + 3)(2x – 1)
(x + 7)
(2x + 3)
12 ~ Algebra I Unit 8
Numerators subtracted over common denominator.
Remember to reverse the signs of each term of the subtrahend.
Subtraction completed, like terms combined.
Numerator and denominator factored, factors canceled.
Simplified answer.
Add or subtract as indicated. 8.4
4
6
6
3
1. 5x + 5x
2. x – 1 + x – 1
4.
6
x 2 + 5x
x+2 + x+2
2x
8
7. x 2 + 8x + 16 + x 2 + 8x + 16
Review
x
3
5. x 2 – 2x – 15 + x 2 – 2x – 15
8.
48 + 4x
2x 2
– 2
x – 36
x – 36
2
9
3x
3. x – 3 – x – 3
5x + 1
2x + 3
6. 6x – 4 – 6x – 4
7x – 2
4 – 3x
9. 2x + 1 + 2x + 1
Solve by using inverse variation. 8.3
10. If n varies inversely as m, and n is 6.25 when m is 8, what is n when m is 40?
11. If b varies inversely as a2, and b is 4 when a is 5, what is b when a is 10?
12. Light intensity varies inversely as the square of the distance from the source of the light. If the
intensity of light is 100 lux at 1 meter from the source, what is the intensity 2 meters away?
Factor by grouping. 7.3
13. 6m + 2mn – 3n – n2
14. x 3 + x 2y 2 – xy – y 3
15. 6s – 36 – rs + 6r
Combine like radicals. If they cannot be combined, write cannot be combined. 7.9
16. 3 √y + 10 √y
17. y √x + 3y √x – 7y √x
18. 5 √8 + 8 √5
Write equations for the lines passing through the set of points. 7.11
19. (– 5, –8) (0, –6)
20. (7, 3) (8, – 5)
21. (–10, –3) (1, 3)
Solve by using direct variation. 7.14
22. If y varies directly as the square of x, and y is 22.5 when x is 3, what is y when x is 5?
23. The distance you are from a lightning strike is directly proportional to the time it takes before you
hear the thunder. If someone 1,372 meters away hears the thunder in 4 seconds, how far away is
someone who doesn’t hear it until 6 seconds after the flash?
24. If b varies directly as a, and b is 7.5 when a is –2.5, what is b when a is 1.2?
8.4 Adding and Subtracting Rationals With Common Denominators ~ 13
Factor polynomials completely. 8.2
25. 3rs 2 + 6rs – 3s 3 – 6s 2
26. 27x 3 + 27x 2 + 6x
Divide the rational expressions. 7.7
28.
6xy – 2x
3x 3y – x 3
÷
4
4y
2xy + 5y
29.
x+3
÷ x2 – 2x – 15
x–5
27. 5 – 5h 8
30.
4x 3y 12x 4y 2
÷
3z 3
9z
Solve for the requested information. 7.12
31. Aaron had two bank accounts, one with $4,000 invested at a 3.5% interest rate, and the other with
$2,700 invested at 4%. Find how long his money was in each account if the money in the $4,000
account was in the bank 18 months longer than the $2,700 investment and his total gain was $582.
Add or subtract as indicated. 8.4
2y 2 – 9y – 9
3y 2 + 5
32. 3xy + 6x – 3xy + 6x
14 ~ Algebra I Unit 8
33.
7x + 4
4x – 3
9x + 21 – 9x + 21
3e 2e + 4
1
34. 15 + 15 + 15