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# 5.E: Review Exercises and Sample Exam

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## Review Exercises

Exercise $$\PageIndex{1}$$ Rules of Exponents

Simplify.

1. $$7^{3}⋅7^{6}$$
2. $$5^{9}5^{6}$$
3. $$y^{5}⋅y^{2}⋅y^{3}$$
4. $$x^{3}y^{2}⋅xy^{3}$$
5. $$−5a^{3}b^{2}c⋅6a^{2}bc^{2}$$
6. $$\frac{55x^{2}yz}{55xyz^{2}}$$
7. $$(\frac{−3a^{2}b^{4}}{2c^{3}})^{2}$$
8. $$(−2a^{3}b^{4}c^{4})^{3}$$
9. $$−5x^{3}y^{0}(z^{2})^{3}⋅2x^{4}(y^{3})^{2}z$$
10. $$(−25x^{6}y^{5}z)^{0}$$
11. Each side of a square measures $$5x^{2}$$ units. Find the area of the square in terms of $$x$$.
12. Each side of a cube measures $$2x^{3}$$ units. Find the volume of the cube in terms of $$x$$.

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.

1. $$8a^{3}−1$$
2. $$5y^{2}−y+1$$
3. $$−12ab^{2}$$
4. $$10$$

1. Binomial; degree $$3$$

3. Monomial; degree $$3$$

Exercise $$\PageIndex{3}$$ Introduction to Polynomials

Write the following polynomials in standard form.

1. $$7−x^{2}−5x$$
2. $$5x^{2}−1−3x+2x^{3}$$

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

Exercise $$\PageIndex{4}$$ Introduction to Polynomials

Evaluate.

1. $$2x^{2}−x+1$$, where $$x=−3$$
2. $$\frac{1}{2}x−\frac{3}{4}$$, where $$x=\frac{1}{3}$$
3. $$b^{2}−4ac$$, where $$a=−\frac{1}{2}, b=−3$$, and $$c=−\frac{3}{2}$$
4. $$a^{2}−b^{2}$$, where $$a=−\frac{1}{2}$$ and $$b=−\frac{1}{3}$$
5. $$a^{3}−b^{3}$$, where $$a=−2$$ and $$b=−1$$
6. $$xy^{2}−2x^{2}y$$, where $$x=−3$$ and $$y=−1$$
7. Given $$f(x)=3x^{2}−5x+2$$, find $$f(−2)$$.
8. Given $$g(x)=x^{3}−x^{2}+x−1$$, find $$g(−1)$$.
9. 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.
10. 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.

1. $$22$$

3. $$6$$

5. $$−7$$

7. $$f(−2)=24$$

9. $$62$$ square units

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

Perform the operations.

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

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.

1. $$6x^{2}(−5x^{4})$$
2. $$3ab^{2}(7a^{2}b)$$
3. $$2y(5y−12)$$
4. $$−3x(3x^{2}−x+2)$$
5. $$x^{2}y(2x^{2}y−5xy^{2}+2)$$
6. $$−4ab(a^{2}−8ab+b^{2})$$
7. $$(x−8)(x+5)$$
8. $$(2y−5)(2y+5)$$
9. $$(3x−1)^{2}$$
10. $$(3x−1)^{3}$$
11. $$(2x−1)(5x^{2}−3x+1)$$
12. $$(x^{2}+3)(x^{3}−2x−1)$$
13. $$(5y+7)^{2}$$
14. $$(y^{2}−1)^{2}$$
15. Find the product of $$x^{2}−1$$ and $$x^{2}+1$$.
16. Find the product of $$32x^{2}y$$ and $$10x−30y+2$$.
17. Given $$f(x)=7x−2$$ and $$g(x)=x^{2}−3x+1$$, find $$(f⋅g)(x)$$.
18. Given $$f(x)=x−5$$ and $$g(x)=x^{2}−9$$, find $$(f⋅g)(x)$$.
19. Given $$f(x)=7x−2$$ and $$g(x)=x^{2}−3x+1$$, find $$(f⋅g)(−1)$$.
20. Given $$f(x)=x−5$$ and $$g(x)=x^{2}−9$$, find $$(f⋅g)(−1)$$.

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.

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

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.

1. $$(−10)^{−2}$$
2. $$−10^{−2}$$
3. $$5x^{−3}$$
4. $$(5x)^{−3}$$
5. $$\frac{1}{7y^{-3}}$$
6. $$3x^{−4}y^{−2}$$
7. $$\frac{−2a^{2}b^{−5}}{c^{−8}}$$
8. $$(−5x^{2}yz^{−1})^{−2}$$
9. $$(−2x^{−3}y^{0}z^{2})^{−3}$$
10. $$(\frac{−10a^{5}b^{3}c^{2}}{5ab^{2}c^{2}})^{−1}$$
11. $$(\frac{a^{2}b^{−4}c^{0}}{2a^{4}b^{−3}c})^{−3}$$

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.

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

2. $$$1,200$$ Exercise $$\PageIndex{10}$$ Negative Exponents Rewrite using scientific notation. 1. $$2,030,000,000$$ 2. $$0.00000004011$$ Answer 2. $$5.796×10^{19}$$ Exercise $$\PageIndex{11}$$ Negative Exponents Perform the indicated operations. 1. $$(5.2×10^{12})(1.8×10^{−3})$$ 2. $$(9.2×10^{−4})(6.3×10^{22})$$ 3. $$\frac{4×10^{16}}{8×10^{−7}}$$ 4. $$\frac{9×10^{−30}}{4×10^{−10}}$$ 5. $$5,000,000,000,000 × 0.0000023$$ 6. $$\frac{0.0003}{120,000,000,000,000}$$ Answer 2. $$5.796×10^{19}$$ 4. $$2.25×10^{−20}$$ 6. $$2.5×10^{−18}$$ ## Simple Exam Exercise $$\PageIndex{12}$$ Simplify. 1. $$−5x^{3}(2x^{2}y)$$ 2. $$(x^{2})^{4}⋅x^{3}⋅x$$ 3. $$\frac{(−2x^{2}y^{3})^{2}}{x^{2}y}$$ 4. 1. $$(−5)^{0}$$ 2. $$−5^{0}$$ Answer 1. $$−10x^{5}y$$ 3. $$4x^{2}y^{5}$$ Exercise $$\PageIndex{13}$$ Evaluate. 1. $$2x^{2}−x+5$$, where $$x=−5$$ 2. $$a^{2}−b^{2}$$, where $$a=4$$ and $$b=−3$$ Answer 1. $$60$$ Exercise $$\PageIndex{14}$$ Perform the operations. 1. $$(3x^{2}−4x+5)+(−7x^{2}+9x−2)$$ 2. $$(8x^{2}−5x+1)−(10x^{2}+2x−1)$$ 3. $$(\frac{3}{5}a−\frac{1}{2})−(\frac{2}{3}a^{2}+\frac{2}{3}a−\frac{2}{9})+(\frac{1}{15}a−\frac{5}{18})$$ 4. $$2x^{2}(2x^{3}−3x^{2}−4x+5)$$ 5. $$(2x−3)(x+5)$$ 6. $$(x−1)^{3}$$ 7. $$\frac{81x^{5}y^{2}z}{-3x^{3}yz}$$ 8. $$\frac{10x^{9}−15x^{5}+5x^{2}}{−5x^{2}}$$ 9. $$\frac{x^{3}−5x^{2}+7x−2}{x−2}$$ 10. $$\frac{6x^{4}−x^{3}−13x^{2}−2x−1}{2x−1}$$ 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. 1. $$2^{−3}$$ 2. $$−5x^{−2}$$ 3. $$(2x^{4}y^{−3}z)^{−2}$$ 4. $$(\frac{−2a^{3}b^{−5}c^{−2}}{ab^{−3}c^{2}})^{−3}$$ 5. Subtract $$5x^{2}y−4xy^{2}+1$$ from $$10x^{2}y−6xy^{2}+2$$. 6. If each side of a cube measures $$4x4$$ units, calculate the volume in terms of $$x$$. 7. 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. 8. 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.
9. 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.
1. $$\frac{1}{8}$$
3. $$\frac{y^{6}}{4x^{8}z^{2}}$$
5. $$5x^{2}y−2xy^{2}+1$$
7. $$118$$ feet
9. $$16.4$$