Algebra

Algebra

Topics

• Similar terms and Simplification

• Equations

• Evaluating algebraic expressions

• Polynomials and Roots

• Operating Polynomials

• Factorizing polynomials

• Rules of Exponents and Radicals

• Sum and Diff of Cubes

• Inequalities

• Functions

• Logarithms

• Matrices

• Quadratics

• Variations

​Elementary

Intermediate

Multiple Choice

1) Points A, B, and C lie on a line in that order, as shown below. The ratio of the length of AB to the length of AC is 7 to 10. What is the ratio of the length of AB to the length of BC, if it can be determined?

• 3:7

• 3:17

• 7:3

• 7:17

• not enough information

Multiple Choice

2) Which of the following expressions is equivalent to

$$\left(3x^2-2x+4\right)+\left(2x-1\right)-\left(x^2-3x+2\right)?$$

• $$x^2-2x+3$$

• $$x^2+2x-1$$

• $$2x^2-3x-5$$

• $$2x^2-3x-1$$

• $$2x^2+3x+1$$

Multiple Choice

3) The following statements are true of the players of a game that uses cards numbered 1–10. I. George has none of the same cards as Ian.

II. George has all the cards that Fiona has.

III. Ian has all the cards that Hailey has.

Which of the following statements must be true?

• George has none of the cards that Fiona has.

• Hailey has none of the cards that Fiona has

• George has all the cards that Hailey has.

• Ian has all the cards that George has

• Hailey has all the cards that Fiona has.

Multiple Choice

4) Which of the following expressions is equivalent to $$\frac{p^2-3p}{2p}+\frac{1}{p^2}?$$

• $$\frac{p^2-3p+1}{2p^3}$$

• $$\frac{2p^2}{p^3-3p^2+2}$$

• $$\frac{p^3-3p^2+2}{2p^2}$$

• $$\frac{p-3p^2}{p^3+4}$$

• $$\frac{2-p^2}{p^2+p-2}$$

Multiple Choice

5) The students in a math class are asked to determine the equation of an unknown linear function, f. The teacher tells the class that the output of the function is 7 when the input is 4. The teacher also says that the output of the function is 28 when the input is 16. Which of the following equations correctly expresses the function f, where x is the input and f(x) is the output?

• $$f\left(x\right)=\frac{4}{7}x+\frac{33}{7}$$

• f(x)=x+12

• f(x)=x+3

• $$f\left(x\right)=\frac{7}{4}x$$

• $$f\left(x\right)=\frac{4}{7}x$$

Getting
Harder

Multiple Choice

6) A salesman earns $600 per week in base salary. For each successful sale, he receives $125 in commission. Which of the following represents the amount of money, in dollars, the salesman earns in a given week in which he makes s successful sales?

• 725 s

• 125s-600

• 600s+125

• 600-125s

• 600+125s

Multiple Choice

7) The expression (n² – 6n + 5)(n + 4) is equivalent to

• A

• B

• C

• D

• E

Multiple Choice

8) If $$f\left(a\right)=a^2+3\ ,\ g\left(a\right)=3a-1,\$$ which of the following is an expression for g(f(a))?

• $$a^2+3a+2$$

• $$-a^2+3a-2$$

• $$3a^2+9a-3$$

• $$3a^2+8$$

• $$3a^2-3$$

Multiple Choice

9) Which of the following expressions is equivalent to (y – 4)^–50 ?

• -50y+200

• $$-y^{50}+4^{50}$$

• $$\frac{1}{\left(-4y\right)^{50}}$$

• $$\frac{1}{\left(y-4\right)^{50}}$$

• $$\frac{1}{y^{50}}-\frac{1}{4^{50}}$$

Multiple Choice

10) In a professional sports league consisting of x teams, y represents the number of teams that qualify for the playoffs in a given season. Which of the following could be used to determine the fraction of teams that does NOT make the playoffs in a given season?

• $$\frac{x-y}{x}$$

• $$\frac{y-x}{x}$$

• $$\frac{x}{y}$$

• $$\frac{x-y}{y}$$

• $$\frac{x+y}{x}$$

​Five More to Go

Multiple Choice

11) For positive real numbers x, y, and z such that 3x = 4y and $$\frac{2}{3}y=\frac{1}{3}z$$ , which of the following inequalities is true?

• x<y<z

• x<z<y

• z<y<x

• y<x<z

• y<z<x

Multiple Choice

12) If x>y, then $$-\left|y-x\right|$$ is equal to?

• x - y

• $$\sqrt[]{y-x}$$

• $$\left|y-x\right|$$

• • $$\left|x-y\right|$$

• • - (x - y)

Multiple Choice

13) What is the LEAST possible integer value of a + b, if 6 < b < a ?

• 9

• 11

• 13

• 14

• 15

Multiple Choice

14) If –1 < x < 0, then which of the following must be true

• $$0^x>\ 0$$

• $$-\frac{1}{x}>1$$

• $$x+\frac{1}{x}=0$$

• $$\frac{1}{x}>0$$

• $$x^0<0$$

Multiple Choice

15) Consider all pairs of positive integers a and b whose sum is 6. For how many values of a does $$a^b=b^a$$ ?

• None

• 1

• 2

• 3

• 6