Download Polynomials

Polynomials
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P.1
How can we rewrite it?
Simplifying Polynomials
P-1
A polynomial is in standard form if the degrees of each term are descending. In other words, the exponents are
decreasing. For example, 8x5 + 13x4 – 2x2 – 11 is in standard form because the exponents are in descending order
(5, 4, 2, 0). Write each polynomial in standard form.
a. 3x – 7x4 + 2x3
b. 11m3 + 2m + 9 – 3m2
P-2
Simplify each polynomial by combining like terms. Write each answer in standard form.
a.
b.
c.
d.
e.
P-3
c. -7y + 2 + y8
d. 5x + 4x2 + 3x3 + 2x4 + x5
-8x + 3 – 2x – 5
4x2 + 3x + 5x2 + 9
7x2 – 8x + 2 – 3x2 – x – 11
4n4 + 2n + 8n2 + 11n + 3n4 + n2
11k5 + 2k + 7k4 – 3k4 + 8k2 – 3k – 4k5
f.
g.
h.
i.
j.
(6x3 + 2x2 + x + 1) + (9x3 + x2 + 11)
(8x4 + 8x + 7) – (3x4 + 2x – 11)
(9g6 – 2g2 – g) + (9g5 + 2g2 + g)
(3x2 + 2x – x2) – (5x2 + x – x2)
(m5 + 3m + m3) + (m2 – 8m5 + m3 – 2m)
For what value of m will each of the following equations be true?
a.
b.
c.
d.
mx3 – 2x + 5x3 + x – 11 = 12x3 – x – 11
(5y8 – 2y4 + y3) + (4y7 + my3 – 2) = 5y8 + 4y7 – 2y4 – y3 – 2
a3 + 2a2 – b5 + 3 + ma3 + b5 – 3 = 2a2
(6n5 – 2n2 + n – 8) – (3n2 – 9n + m) = 6n5 – 5n2 + 10n – 13
Review & Preview
P-4
P-7
Find the following products.
a. 3x ∙ 7x
b. 11x2 ∙ 2x
c. 9 ∙ 3x5
d. 7y3 ∙ 3y2
e.
f.
g.
h.
-8m5 ∙ m
6h8 ∙ 8h6
-1.5p ∙ -2p7
13g8 ∙ -4
i.
j.
k.
l.
j11 ∙ j12
9g4h2 ∙ -8g2h5
3x2y ∙ 2xy2
-7a3 ∙ 2b2
There are already 171/2 gallons of water in Martha’s bathtub. She turns on the faucet and it releases water at a rate
of 61/4 gallons per minute.
a. Write an equation relating the total gallons of water in the tub, y, to how long the faucet is on in minutes,
x.
b. Her tub holds 80 gallons of water. How long will it take her to fill the tub?
c. Martha’s boyfriend Jake also has an 80 gallon tub. However, his faucet releases water twice as fast as
Martha’s. If he starts filling his tub when it has 51/2 gallons of water already in it, how long will it take
his tub to be full? Round your answer to the nearest minute.
P-8
Multiple Choice. Jarrell’s parents told him he could spend up to $20 on balloons and party hats for his birthday
party. Each pack of balloons cost $2.50 and each pack of party hats cost $1.75. Jarrell can use the following
inequality to determine the possible combinations of balloons and party hats he can buy, where x represents the
number of packs of balloons and y represents the number of packs of party hats.
2.50x + 1.75y < 20.00
If y = 3, which of the following statements is true.
a.
b.
c.
d.
Jarrell can purchase 6 packs of balloons.
Jarrell can purchase a maximum of 3 packs of balloons.
Jarrell can purchase a minimum of 3 packs of balloons.
Jarrell can purchase 3 packs of balloons, but this is not the minimum or maximum.
P-9
Multiple Choice. Ukani found an online store that sells scarves. Every scarf costs $12 and there is a flat rate for
shipping. Ukani bought 7 scarves and spent a total $93.00. Which equation can be used to find the cost of
shipping?
a. 93 = 7x + 12
b. 93 = 12x + 7
P-10
c. 93 = 12(7) + x
d. 93 = 84x
Solve the following system of equations using any technique
5x + y = 28
2x – 3y = 18
P-11
Graph the linear inequality. Plot and verify at least one solution.
5x – y < 10
P.2
How can we rewrite it?
Simplifying polynomials with the distributive property
P-12
P-14
P-15
Simplify each expression using the distributive property.
a. 3(2x + 9)
b. -4(3y – 1)
c. 7x(5x – 4y)
d. –y(11 – y)
Simplify each expression
a. 6x3 • 4x5
b. 6x3 • -2x4
c. 6x3 • x2
d. 6x3 • -7
Consider the following problem:
Simplify 6x3(4x5 – 2x4 + x2 – 7) using the distributive property.
Explain how this problem is similar to the previous problem.
P-16
Simplify each polynomial. Write each answer in standard form.
a. 4x5(2x2 – 5)
b. b3(8b2 – 11b + 2)
c. 10x(x2 – 4x + 4)
P-17
d. -2y(y7 – 2y3 + y + 1)
e. 3g4(5g2 + 3) + 2g(7g3 – 1)
f. -8(a3 – 2b2 + 1) – 3a2(-2a) + 5a(a2 + 1)
Fill in the blank to make the equation true.
a. ___(7x – 3) = -14x + 6
b. ___(y2 – 2y + 1) = 7y2 – 14y + 7
c. ___(x2 + 11x + 24) = x3 + 11x2 + 24x
d. ___(4x3 + 2x2 + x + 1) = 4x5 + 2x4 + x3 + x2
e.
f.
g.
h.
___(5n5 – 3) = 15n8 – 9n3
___(x2 + 7x + 6) = 5x4 + 35x3 + 30x2
___(25y2 – 16) = 250y3 – 160y
___(4b2 – 1) = 8ab2 – 2a
Review & Preview
P-18
Multiple Choice. For what value of m will the following equation be true?
mx(4x3 – 2x2) + 6x4 = -2x4 + 4x3
a. m = -2
P-19
c. m = 1/2
d. m = 2
Solve each inequality and represent the solution on a number line.
a. 3x + 1 < 11
P-20
b. m = -1/2
b. 1.5 – 2x > 0.3x – 1.375
c. |x + 4| < 9
Multiple Choice. Mr. Patlikov and his model UN club are trying to raise money for a field trip to New York City
to visit the United Nations Headquarters. They purchased subs from a fundraising company, and they will resell
them for a profit. The following equation can be used to determine the profit, y, based on the number of subs
sold, x.
y = 5.50x – 165
Roland, the club president, correctly determined that the x-intercept is (30, 0). Which of the following statements
is true?
a.
b.
c.
d.
The club made a profit of $30.
The club spent $30 buying the subs from the fundraising company.
The club needs to sell 30 subs to break even.
The club is selling their subs for $30 each.
P-21
Factor each expression completely
a. x2 + 9x – 22
P-22
b. x2 – 14x + 24
c. 7x3 + 21x2 + 14x
Multiple Choice. Kiran and her neighbor Lucia wants to plant tomatoes and pepper plants in her garden. Kiran
buys 4 tomato plants each for x dollars and 2 pepper plants each for y dollars and spent $20. Lucia buys 1 pepper
plant for y dollars and 6 tomato plants each for x dollars and spent $24. The following system below represents
this situation.
4x + 2y = 20
6x + y = 24
Which of the following statements is true?
a.
b.
c.
d.
P-24
Tomato plants cost $4 or $6.
Kiran spent $14 on tomato plants.
Pepper plants are more expensive than tomato plants.
Pepper plants cost $3.50 each.
Solve each equation. Be sure to find all solutions. Round any irrational number to the nearest hundredth.
a.
b.
c.
d.
8x – 3 = 2(x + 11)
2(g – 2) = 5g – 11 – 3g + 7
x2 = 125
x3 = 125
e.
f.
g.
h.
|x| = 125
x2 = -125
x3 = -125
|x|= -125
P.3
How can I rewrite it?
Factoring Polynomials
P-25
Factor each trinomial
a. x2 – 5x – 36
P-26
b. 2x2 + 9x + 10
Earlier in the year, we learned how to factor expressions by first looking for a common factor.
For example: Factor 5x3 + 45x2 – 50x, we first factored out the common factor 5x before trying to make a generic
rectangle.
5x
5x3
x2
+45x2
+9x
-50x
-10
= 5x(x2 + 9x – 10)
For each polynomial below, factor out the greatest common factor. Do nothing more for this problem. The
answer will look like the last step in the example above.
a.
b.
c.
d.
P-27
10a2 – 30a – 40
x3 + 11x2 + 3x
9y2 – 900
-3z3 – 12z2 – 12z
e.
f.
g.
h.
15x4 – 15x2
5g5 + 2g3 + g2
x6 – 20x5 + 10x4
18ax2 – 3a
FACTORING COMPLETELY. To factor an expression completely, you must first factor out the greatest
common factor if there is one, and then factor the remaining polynomial if possible (using a generic rectangle and
diamond problem).
For example: In the previous problem, when we factored out the common factor of 5x3 + 45x2 – 50x we got
5x(x2 + 9x – 10). Now we look at the remaining polynomial x2 + 9x – 10. There is no more common factor
(because 5x was the greatest), so we use a generic rectangle and diamond problem to see if we can factor it that
way.
-10
-10x2
-x
x2
10x
9x
-1
-x
-10
x
x2
10x
x
10
So, 5x3 + 45x2 – 50x = 5x(x – 1)(x + 10)
(You can’t forget the common factor, 5x, in the answer)
Factor each polynomial completely. (If there is no common factor, go straight to the generic rectangle. If after
factoring out the greatest common factor the remaining polynomial cannot be factored further, then you are done
and write your answer like in problem [P-26].)
a. 10x2 + 80x + 120
b. x3 – 36x
c. -3x3 – 12x2 – 12x
d. 7ax2 – 21ax – 70a
e. 2x2 + 11x + 14
f. 4x2 + 8x
Review & Preview
P-28
Multiple Choice. Which point needs to be removed from the following relation so that the resulting relation
would be a function?
x
y
-5
3
2
12
6
-1.5
3
3
6
-1.5
a. (3, 3)
b. (6, -1.5)
P-29
2
-12
-9
2
c. (2, -12)
d. (-9, 2 )
Represent the solution to the system of inequalities on a graph.
y < -1/2 x + 3
2x – 3y > 0
P-30
Multiple Choice. When the following expression is simplified, what is the coefficient of the x-term?
(2x + 3)(4x2 – 2x + 11)
a. -6
P-31
P-32
b. 1
c. 16
d. 22
Solve each equation.
a.
x=9
c.
x  11 = 5
b.
2x = 4
d.
21x = 7 3
Multiple Choice. Angle C is a right angle. Which statement is true about the hypotenuse?
A
18
C
80
a. The hypotenuse equals
6724 and is a rational number.
b. The hypotenuse equals
6724 and is an irrational number.
B
98 and is a rational number.
d. The hypotenuse equals 98 and is an irrational number.
c. The hypotenuse equals
P-33
Solve the system of equations using any technique
5x + y = 11
y = 2x – 3
P-34
Multiple Choice. Line m goes through the points (-2, 6) and (0, 9). Line n goes through the points (5, 0) and
(11, q). What must be the value of q so that line n is perpendicular to line m?
a. -9
b. -4
c. 4
d. 9
P.4
How can we find it?
Polynomials and areas
P-35
Simplify each product. Write each answer in standard form.
a. 4x • 3x
b. w(w – 4)
P-36
e. (y + 3)(y2 + 6y – 11)
f. 5x2(3x3 – 2x + 5)
c. (x + 4)(2x + 1)
d. (h – 3)(h + 3)
To find the area of a rectangle, we simply use the formula A = lw (area = length • width). This does not change
even if the lengths and the widths are polynomials. Find the area of each rectangle.
a.
d.
4.5
x+3
x–4
8
b.
e.
h
4.5x
2h + 1
f.
8x
c.
3x – 1
2
3x – 1
w+3
P-37
Amal took a picture to a printing center. She wants the length of her picture to be 8 inches longer than the height,
as shown in the diagram below.
h
h+8
a. Write a simplified polynomial that represents the area of this picture.
Amal then adds a picture frame that is 2-inches wide on all sides, as shown in the diagram below.
2
h + ___
2
h
2
h+8
2
h + ___
b. Fill in the blanks with numbers to represent the total length and height of the picture and frame.
c. What is the total area of the picture and frame?
d. To find the area of the picture frame alone, Amal knows she would have to subtract the area of just the
picture from the total area. Which of the two expressions below would find that area? Then, simplify
that expression.
i. h2 + 16h + 48 – h2 + 8h
ii. h2 + 16h + 48 – (h2 + 8h)
P-38
Le’von needs help constructing blueprints for a rectangular in-ground pool. He wants the length of his pool to be
5 feet longer than twice the width. Let w equal the width of the pool.
a. Draw and label a diagram in terms of w of the pool and walkway.
b. Find the area of the pool.
He also needs a four-foot-wide walkway around the pool on all sides.
c.
d.
e.
f.
P-39
Add this to your diagram.
Find the total area of the pool and walkway.
Find the area of the walkway alone.
Le’von plans on using rectangular bricks for the walkway around the pool. He has enough bricks to cover
536 ft2. What should you tell him to make the width of his pool?
Chijindum panted on a canvas that is 6 inches wider than it is tall. He then framed his painting, and the total area
of the picture and frame is represented by the polynomial x2 + 10x + 16, where x equals the height of the canvas.
If the frame is the same width around each side, how wide is the frame?
Review & Preview
P-40
Simplify each expression
a. 8x4 • -1.5x7
c. (-4a3b)2
8
b.
P-41
 m7
d.  9
m
3y
6y



3
Multiple Choice. The linear equation y = 1.17x – 2.05 can approximate the height of a white oak tree in feet, y,
based on the number of years since the seed was planted, x.
Which statement is true?
a.
b.
c.
d.
P-42
A white oak tree is approximately 2.05 feet tall in 1.17 years.
A white oak tree grows approximately 1 feet 17 inches each year.
A white oak tree will be approximately 9.65 feet tall after 10 years.
A white oak tree shrinks approximately 2.05 feet every year.
Solve the system of equations using any technique.
a + s = 724
8a + 5s = 4556
P-43
Multiple Choice. Which of the following expressions is a factor of the polynomial given below?
10x3 – 50x2 + 60x
a. 10x3
b. (x – 3)
c. (x + 2)
d. (10x – 3)
P-44
What is the equation of the line through the points (1.25, 8) and (-2, 6.7)?
P-45
Multiple Choice. Which value must be removed so that the new mean (average) of the data set would be 79?
57, 84, 72, 66, 77, 89, 75, 68, 88, 100, 91, 93, 60, 73, 73
a. 57
b. 60
(Problem P-46 is on the next page)
c. 77
d. 100
P-46
A marble is rolling down a ramp, as shown in the diagram.
15 in
If this ramp has a slope of -2/3, what is the diagonal length that the marble rolls? Round to the nearest hundredth if
needed.
P.5
How can we rewrite it?
Simplifying rational expressions
P-47
Factor each expression completely.
a. x2 + 7x + 12
P-48
a.
b.
c.
d.
P-49
a.
P-50
a.
P-51
b. -3x2 – 18x – 27
c. 8y3 – 40y2 – 48y
When simplifying rational expressions, we learned to look for 1’s to divide out. Simplify each rational expression
below.
x3
x3
(a  4)(a  2)
(a  4)(a  11)
(3g  4) g
g (3g  4)
(n  3)(n  4)(n  5)
(n  11)(n  3)(n  4)
e.
f.
g.
h.
xxx y yzzzz
x y y y y
(b  4) 2
(b  4)(b  1)
2h  1
3h(2h  1)
3h(2h  1)
2h  1
Remember, in order to simplify rational expressions by dividing out 1’s, we must first make sure both the
numerator and denominator are written as a product by factoring completely if needed. Simplify each of the
rational expressions below.
x 2  6x  8
( x  2)( x  8)
b.
x( x  3)( x  1)
x 3  2x 2  x
c.
Simplify the following rational expressions by first simplifying the numerator and denominator independently, if
possible, and then factoring and simplifying like the previous problems.
x( x  7)  12
3x  x  2 x 2  20
2
b.
3( x  4)  ( x 2  15 x  16)
x(2 x  4)  x( x  2)
Ask your teacher for the accompanying cards for this activity. Align the sides of an expression and its simplified
form to make a 5-by-5 square.
Review & Preview
P-52
Multiple Choice. What is the product of 5(x – 3)(2x + 1)?
a. 15x – 10
b. 10x2 – 25x – 15
P-53
 5 x 2  125
5 x 2  25 x
c. 50x2 – 125x – 75
d. 50x2 – 25x – 15
Solve each equation or inequality. Represent the solution(s) on a number line.
a. (x – 1)2 = 16
b. |9 – x| < 3
c. 2(7x – 3) = 5(x + 1)
d. x2 > 9
P-54
Multiple Choice. Approximate the equation of the line of best fit.
a. y = -x + 100
b. y = -9.5x – 100
P-55
c. y = 9.5x + 100
d. y = -9.5x + 100
The path of a water balloon toss is given by the quadratic equation y = -x2 + 18x – 32, where x represents the
distance in yards from the goal line on a football field, and y represents the height of the water balloon in yards.
a. What is the horizontal distance the water balloon traveled?
b. How high did the water balloon reach?
P-56
Multiple Choice. What is the slope of the line through the points (2, 9) and (2, -1)?
a. 0
b. 10
c. -10
d. Undefined
P-57
Eliza went to the store to buy chips and soda for a little party she is having with her friends. Her foster mom told
her she could spend less than $36 and Eliza wants to buy at least 4 bags of chips and at least 3 cases of soda.
Each bag of chips costs $3.00 and each case of soda costs $4.50. Represents this situation with a system of
inequalities where x represents the number of bags of chips and y represents the number of cases of soda. Then,
represent the solutions to this system on a graph.
P-58
Multiple Choice. Which data set has a median and mean that are equal?
a. 12, 5, 18, 10, 10, 13
b. 2, 14, 8, 8, 8, 10, 13, 15
c. 6, 12, 4, 10, 18
d. 1, 6, 6, 1, 6, 8, 10