Algebra Problem Set
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1. Decide whether each of the following is a quadratic equation or not.
a. 3 - X^2 = 9x
b. X^3 - 8 = 0
c. 2X^2 - 7X + 1
d. -5 - X^2 + 8X = 0
e. 6 + 5X - X^2
2. Find f(-3) if f(x) = -2x^3-8x^2+2x-1
3. Do the math and express without negative exponents: x^(-7) * x^3
4. Do the math and express without negative exponents: x^7 / x^3
5. Do the math and express without negative exponents: (x^7)^(-3)
6. In January of 1990, you deposited $1000 into an account that guaranteed an annual interest rate of 3%. You didn't touch your money for 17 years, and in January 2007, you took out all your money. How much did you have? Round to the nearest cent.
7. Express in the OTHER notation (if it's in scientific notation, re-express in regular notation; if it's in regular notation, change it to scientific notation):
a. 5,900,000
b. 0.00082
c. 5 x 10^(-6)
d. 8.1 x 10^7
e. 2 E-08
8. Multiply: (4X^3 - 7X^2 + X - 2) * (3X^2 - 2X + 3)
9. Divide: (4X^3 - 7X^2 + X - 2) / (X + 6)
10. Do the math: (3X - 2)^2
11. Do the math: (3X - 2)*(3X + 2)
12. Factor completely: 5X^2+5X-30
13. Factor completely: 8X^2+2X-3
14. Factor completely: 4X^2 - 81
15. The length of a rectangular field is (3X^2 - 2X - 8) feet. The width is (X^2 + 4X + 1) feet. Write the polynomial that represents the AREA of the field. Be sure to use correct units.
16. An object is thrown upward at an initial velocity of 72 feet per second, from a platform 40 feet high. The function that describes the object's height in feet above the ground t seconds after its release is as follows:
h(t) = -16t^2 + 72t + 40
The -16 times t^2 is due to gravity, on planet Earth, and using feet and seconds as our units of measure. The 72 times t is due to the initial velocity of the object. The constant term 40 is due to the initial height of the object.
Find h(3.5) and interpret it. Be sure to use correct units.
17. See #16. When will the object hit the ground? (When will the height above the ground be exactly 0 feet?) Be sure to use correct units.
18. Solve: X^2 + 5X - 24 = 0
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Solution Summary
This problem set has 18 questions involving quadratic equations, exponents, interest rates, scientific notation, multiplication and division of polynomials, and word problems.
Solution Preview
1. Decide whether each of the following is a quadratic equation or not.
a. 3 - X^2 = 9x
b. X^3 - 8 = 0
c. 2X^2 - 7X + 1
d. -5 - X^2 + 8X = 0
e. 6 + 5X - X^2
Quadratic equations have an x^2 (x-squared) term as the largest term. Therefore, a, c, d, and e are quadratic equations, and b is not one, since it has an x^3 term.
2. Find f(-3) if f(x) = -2x^3-8x^2+2x-1
Just plug -3 in for x:
f(x) = -2x^3-8x^2+2x-1
f(-3) = -2(-3)^3 - 8(-3)^2 + 2(-3) - 1
f(-3) = -2*-27 - 8*9 + 2(-3) - 1
f(-3) = 54 - 72 - 6 - 1
f(-3) = -25
3. Do the math and express without negative exponents: x^(-7) * x^3
You can change negative exponents to positive exponents by moving the term to the bottom of the fraction (or to the top, if the term was originally at the bottom). You can combine exponents by adding them (if the two terms are being multiplied, like in this problem), by subtracting them (if the two terms are being divided, like in problem 4), or by multiplying them (if one exponent is being raised to another, like in problem 5).
x^(-7) * x^3
x^(-7 + 3)
x^(-4)
1/x^4
The answer is 1/(x^4). This is the same as (1/x)^4, so you can write it either way.
4. Do the math and express without negative exponents: x^7 / x^3
x^7 / x^3
x^(7 - 3)
x^(4)
5. Do the math and ...
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