5.E: Review Exercises and Sample Exam

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Review Exercises

Exercise $\PageIndex{1}$ Rules of Exponents

Simplify.

$7^{3}⋅7^{6}$
$5^{9}5^{6}$
$y^{5}⋅y^{2}⋅y^{3}$
$x^{3}y^{2}⋅xy^{3}$
$−5a^{3}b^{2}c⋅6a^{2}bc^{2}$
$\frac{55x^{2}yz}{55xyz^{2}}$
$(\frac{−3a^{2}b^{4}}{2c^{3}})^{2}$
$(−2a^{3}b^{4}c^{4})^{3}$
$−5x^{3}y^{0}(z^{2})^{3}⋅2x^{4}(y^{3})^{2}z$
$(−25x^{6}y^{5}z)^{0}$
Each side of a square measures $5x^{2}$ units. Find the area of the square in terms of $x$.
Each side of a cube measures $2x^{3}$ units. Find the volume of the cube in terms of $x$.

Answer

1. $7^{9}$

3. $y^{10}$

5. $−30a^{5}b^{3}c^{3}$

7. $\frac{9a^{4}b^{8}}{4c^{6}}$

9. $−10x^{7}y^{6}z^{7}$

11. $A=25x^{4}$

Exercise $\PageIndex{2}$ Introduction to Polynomials

Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

$8a^{3}−1$
$5y^{2}−y+1$
$−12ab^{2}$
$10$

Answer

1. Binomial; degree $3$

3. Monomial; degree $3$

Exercise $\PageIndex{3}$ Introduction to Polynomials

Write the following polynomials in standard form.

$7−x^{2}−5x$
$5x^{2}−1−3x+2x^{3}$

Answer: 1. $-x^{2}-5x+7$

Exercise $\PageIndex{4}$ Introduction to Polynomials

Evaluate.

$2x^{2}−x+1$, where $x=−3$
$\frac{1}{2}x−\frac{3}{4}$, where $x=\frac{1}{3}$
$b^{2}−4ac$, where $a=−\frac{1}{2}, b=−3$, and $c=−\frac{3}{2}$
$a^{2}−b^{2}$, where $a=−\frac{1}{2}$ and $b=−\frac{1}{3}$
$a^{3}−b^{3}$, where $a=−2$ and $b=−1$
$xy^{2}−2x^{2}y$, where $x=−3$ and $y=−1$
Given $f(x)=3x^{2}−5x+2$, find $f(−2)$.
Given $g(x)=x^{3}−x^{2}+x−1$, find $g(−1)$.
The surface area of a rectangular solid is given by the formula $SA=2lw+2wh+2lh$, where $l, w$, and $h$ represent the length, width, and height, respectively. If the length of a rectangular solid measures $2$ units, the width measures $3$ units, and the height measures $5$ units, then calculate the surface area.
The surface area of a sphere is given by the formula $SA=4πr^{2}$, where $r$ represents the radius of the sphere. If a sphere has a radius of $5$ units, then calculate the surface area.

Answer

1. $22$

3. $6$

5. $−7$

7. $f(−2)=24$

9. $62$ square units

Exercise $\PageIndex{5}$ Adding and Subtracting Polynomials

Perform the operations.

$(3x−4)+(9x−1)$
$(13x−19)+(16x+12)$
$(7x^{2}−x+9)+(x^{2}−5x+6)$
$(6x^{2}y−5xy^{2}−3)+(−2x^{2}y+3xy^{2}+1)$
$(4y+7)−(6y−2)+(10y−1)$
$(5y^{2}−3y+1)−(8y^{2}+6y−11)$
$(7x^{2}y^{2}−3xy+6)−(6x^{2}y^{2}+2xy−1)$
$(a^{3}−b^{3})−(a^{3}+1)−(b^{3}−1)$
$(x^{5}−x^{3}+x−1)−(x^{4}−x^{2}+5)$
$(5x^{3}−4x^{2}+x−3)−(5x^{3}−3)+(4x^{2}−x)$
Subtract $2x−1$ from $9x+8$.
Subtract $3x^{2}−10x−2$ from $5x^{2}+x−5$.
Given $f(x)=3x^{2}−x+5$ and $g(x)=x^{2}−9$, find $(f+g)(x)$.
Given $f(x)=3x^{2}−x+5$ and $g(x)=x^{2}−9$, find $(f−g)(x)$.
Given $f(x)=3x^{2}−x+5$ and $g(x)=x^{2}−9$, find $(f+g)(−2)$.
Given $f(x)=3x^{2}−x+5$ and $g(x)=x^{2}−9$, find $(f−g)(−2)$.

Answer

1. $12x−5$

3. $8x^{2}−6x+15$

5. $8y+8$

7. $x^{2}y^{2}−5xy+7$

9. $x^{5}−x^{4}−x^{3}+x^{2}+x−6$

11. $7x+9$

13. $(f+g)(x)=4x^{2}−x−4$

15. $(f+g)(−2)=14$

Exercise $\PageIndex{6}$ Multiplying Polynomials

Multiply.

$6x^{2}(−5x^{4})$
$3ab^{2}(7a^{2}b)$
$2y(5y−12)$
$−3x(3x^{2}−x+2)$
$x^{2}y(2x^{2}y−5xy^{2}+2)$
$−4ab(a^{2}−8ab+b^{2})$
$(x−8)(x+5)$
$(2y−5)(2y+5)$
$(3x−1)^{2}$
$(3x−1)^{3}$
$(2x−1)(5x^{2}−3x+1)$
$(x^{2}+3)(x^{3}−2x−1)$
$(5y+7)^{2}$
$(y^{2}−1)^{2}$
Find the product of $x^{2}−1$ and $x^{2}+1$.
Find the product of $32x^{2}y$ and $10x−30y+2$.
Given $f(x)=7x−2$ and $g(x)=x^{2}−3x+1$, find $(f⋅g)(x)$.
Given $f(x)=x−5$ and $g(x)=x^{2}−9$, find $(f⋅g)(x)$.
Given $f(x)=7x−2$ and $g(x)=x^{2}−3x+1$, find $(f⋅g)(−1)$.
Given $f(x)=x−5$ and $g(x)=x^{2}−9$, find $(f⋅g)(−1)$.

Answer

1. $−30x^{6}$

3. $10y^{2}−24y$

5. $2x^{4}y^{2}−5x^{3}y^{3}+2x^{2}y$

7. $x^{2}−3x−40$

9. $9x^{2}−6x+1$

11. $10x^{3}−11x^{2}+5x−1$

13. $25y^{2}+70y+49$

15. $x^{4}−1$

17. $(f⋅g)(x)=7x^{3}−23x^{2}+13x−2$

19. $(f⋅g)(−1)=−45$

Exercise $\PageIndex{7}$ Dividing Polynomials

Divide.

$\frac{7y^{2}−14y+28}{7}$
$\frac{12x^{5}−30x^{3}+6x}{6x}$
$\frac{4a^{2}b−16ab^{2}−4ab}{−4ab}$
$\frac{6a^{6}−24a^{4}+5a^{2}}{3a^{2}}$
$(10x^{2}−19x+6)÷(2x−3)$
$(2x^{3}−5x^{2}+5x−6)÷(x−2) $
$\frac{10x^{4}−21x^{3}−16x^{2}+23x−20}{2x−5}$
$\frac{x^{5}−3x^{4}−28x^{3}+61x^{2}−12x+36}{x−6}$
$\frac{10x^{3}−55x^{2}+72x−4}{2x−7}$
$\frac{3x^{4}+19x^{3}+3x^{2}−16x−11}{3x+1}$
$\frac{5x^{4}+4x^{3}−5x^{2}+21x+21}{5x+4}$
$\frac{x^{4}−4}{x−4}$
$\frac{2x^{4}+10x^{3}−23x^{2}−15x+30}{2x^{2}−3}$
$\frac{7x^{4}−17x^{3}+17x^{2}−11x+2}{x^{2}−2x+1}$
Given $f(x)=x^{3}−4x+1$ and $g(x)=x−1$, find $(f/g)(x)$.
Given $f(x)=x^{5}−32$ and $g(x)=x−2$, find $(f/g)(x)$.
Given $f(x)=x^{3}−4x+1$ and $g(x)=x−1$, find $(f/g)(2)$.
Given $f(x)=x^{5}−32$ and $g(x)=x−2$, find $(f/g)(0)$.

Answer

1. $y^{2}−2y+4$

3. $−a+4b+1$

5. $5x−2$

7. $5x^{3}+2x^{2}−3x+4$

9. $5x^{2}−10x+1+\frac{3}{2x−7}$

11. $x^{3}−x+5+\frac{1}{5x+4}$

13. $x^{2}+5x−10$

15. $(f/g)(x)=x^{2}+x−3−\frac{2}{x−1}$

17. $(f/g)(2)=1$

Exercise $\PageIndex{8}$ Negative Exponents

Simplify.

$(−10)^{−2}$
$−10^{−2}$
$5x^{−3}$
$(5x)^{−3}$
$\frac{1}{7y^{-3}}$
$3x^{−4}y^{−2}$
$\frac{−2a^{2}b^{−5}}{c^{−8}}$
$(−5x^{2}yz^{−1})^{−2}$
$(−2x^{−3}y^{0}z^{2})^{−3}$
$(\frac{−10a^{5}b^{3}c^{2}}{5ab^{2}c^{2}})^{−1}$
$(\frac{a^{2}b^{−4}c^{0}}{2a^{4}b^{−3}c})^{−3}$

Answer

1. $\frac{1}{100}$

3. $\frac{5}{x^{3}}$

5. $\frac{y^{3}}{7}$

7. $\frac{−2a^{2}c^{8}}{b^{5}}$

9. $\frac{−x^{9}}{8z^{6}}$

11. $8a^{6}b^{3}c^{3}$

Exercise $\PageIndex{9}$ Negative Exponents

The value in dollars of a new laptop computer can be estimated by using the formula $V=1200(t+1)^{−1}$, where $t$ represents the number of years after the purchase.

Estimate the value of the laptop when it is $1\frac{1}{2}$ years old.
What was the laptop worth new?

Answer: 2. $$1,200$

Exercise $\PageIndex{10}$ Negative Exponents

Rewrite using scientific notation.

$2,030,000,000$
$0.00000004011$

Answer: 2. $5.796×10^{19}$

Exercise $\PageIndex{11}$ Negative Exponents

Perform the indicated operations.

$(5.2×10^{12})(1.8×10^{−3})$
$(9.2×10^{−4})(6.3×10^{22})$
$\frac{4×10^{16}}{8×10^{−7}}$
$\frac{9×10^{−30}}{4×10^{−10}}$
$5,000,000,000,000 × 0.0000023$
$\frac{0.0003}{120,000,000,000,000}$

Answer

2. $5.796×10^{19}$

4. $2.25×10^{−20}$

6. $2.5×10^{−18}$

Simple Exam

Exercise $\PageIndex{12}$

Simplify.

$−5x^{3}(2x^{2}y)$
$(x^{2})^{4}⋅x^{3}⋅x$
$\frac{(−2x^{2}y^{3})^{2}}{x^{2}y}$
1. $(−5)^{0}$
2. $−5^{0}$

Answer

1. $−10x^{5}y$

3. $4x^{2}y^{5}$

Exercise $\PageIndex{13}$

Evaluate.

$2x^{2}−x+5$, where $x=−5$
$a^{2}−b^{2}$, where $a=4$ and $b=−3$

Answer: 1. $60$

Exercise $\PageIndex{14}$

Perform the operations.

$(3x^{2}−4x+5)+(−7x^{2}+9x−2) $
$(8x^{2}−5x+1)−(10x^{2}+2x−1) $
$(\frac{3}{5}a−\frac{1}{2})−(\frac{2}{3}a^{2}+\frac{2}{3}a−\frac{2}{9})+(\frac{1}{15}a−\frac{5}{18})$
$2x^{2}(2x^{3}−3x^{2}−4x+5)$
$(2x−3)(x+5)$
$(x−1)^{3}$
$\frac{81x^{5}y^{2}z}{-3x^{3}yz}$
$\frac{10x^{9}−15x^{5}+5x^{2}}{−5x^{2}}$
$\frac{x^{3}−5x^{2}+7x−2}{x−2}$
$\frac{6x^{4}−x^{3}−13x^{2}−2x−1}{2x−1}$

Answer

1. $−4x^{2}+5x+3 $

3. $−\frac{2}{3}a^{2}−\frac{5}{9}$

5. $2x^{2}+7x−15 $

7. $−27x^{2}y$

9. $x^{2}−3x+1$

Exercise $\PageIndex{15}$

Simplify.

$2^{−3}$
$−5x^{−2}$
$(2x^{4}y^{−3}z)^{−2}$
$(\frac{−2a^{3}b^{−5}c^{−2}}{ab^{−3}c^{2}})^{−3}$
Subtract $5x^{2}y−4xy^{2}+1$ from $10x^{2}y−6xy^{2}+2$.
If each side of a cube measures $4x4$ units, calculate the volume in terms of $x$.
The height of a projectile in feet is given by the formula $h=−16t^{2}+96t+10$, where $t$ represents time in seconds. Calculate the height of the projectile at $1\frac{1}{2}$ seconds.
The cost in dollars of producing custom t-shirts is given by the formula $C=120+3.50x$, where $x$ represents the number of t-shirts produced. The revenue generated by selling the t-shirts for $$6.50$ each is given by the formula $R=6.50x$, where $x$ represents the number of t-shirts sold.
1. Find a formula for the profit. (profit = revenue − cost)
2. Use the formula to calculate the profit from producing and selling $150$ t-shirts.
The total volume of water in earth’s oceans, seas, and bays is estimated to be $4.73×10^{19}$ cubic feet. By what factor is the volume of the moon, $7.76×10^{20}$ cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth.

Answer

1. $\frac{1}{8}$

3. $\frac{y^{6}}{4x^{8}z^{2}}$

5. $5x^{2}y−2xy^{2}+1$

7. $118$ feet

9. $16.4$