Question 1

Simplify the expression $$ (2$$^{-3}$$y)^2$$ $$(6^{-1}y$$^{-2}$$) $$ and write your answer without negative exponents.

Accepted Answer

$$ \frac{1}{36y^2$} $$

Question 2

Which of the following is the correct simplification of $$ \frac{2$$^{-1}$$x^{-3}y^0$}{3^3$x^{-5}y$$^{-2}$$} $$?

Accepted Answer

$$ \frac{1}{27x^2$y^2$$} $

Question 3

A chemist is mixing two solutions. The first solution is represented by $$ (9r + 7s - t) $$ and the second by $$ (-2r - 2s - 3t) . $$What is the simplified sum of the two solutions?

Accepted Answer

$$ 7r + 5s - 4t $$

Question 4

Expand and simplify $$ (2 - 3x^2$ + x)(5x - 3) .$$

Accepted Answer

$$ 5x^2$ - 15x^3$ + 2x - 6 $$

Question 5

Divide using long division: $$ (4x^2$ - 12x + 3x^5$ - 17) \div (-2 + x^2$) . $$What is the leading term of the quotient?

Accepted Answer

$$ 3x^3$ $$

Question 6

Use synthetic division to divide $$ (x^4$ - 3x^3$ + 3x^2$ + 2x)  by$$ $$ (x - 2) . $$What is the remainder?

Accepted Answer

$$ 8 $$

Question 7

Using the Remainder Theorem, determine whether $$ x + 3  is a $$factor of $$ -2x^2$ + 3x - 2 .$$

Accepted Answer

No, because the remainder is $$ -11 .$$

Question 8

Factor completely: $$ 4x^3$ + 12x^2$ - x - 3 . $$Which of the following is a correct factorization?

Accepted Answer

$$ (x + 3)(4x^2$ - x - 1) $$

Question 9

Solve by factoring: $$ 15x^2$ = 3x .$$

Accepted Answer

$$ x = 0, x = 5 $$

Question 10

Solve using the quadratic formula: $$ x^2$ + 9x + 18 = 0 .$$

Accepted Answer

$$ x = -3, x = -6 $$

Question 11

A rectangular garden has an area represented by $$ 4x^2$ + 17x - 15 . $$Which of the following represents its possible dimensions (factored form)?

Accepted Answer

$$ (4x - 3)(x + 5) $$

Question 12

A company’s profit is modeled by $$ 12h^4$ - h^2$ - 35 . $$Factor this expression completely.

Accepted Answer

$$ (3h^2$ - 7)(4h^2$ + 5) $$

Question 13

Solve by factoring: $$ x(x + 3) = 4 .$$

Accepted Answer

$$ x = -4, x = 1 $$

Question 14

A projectile's height is modeled by $$ c^2$ = 5c . $$Solve for $$ c $$ using the quadratic formula.

Accepted Answer

$$ c = 0, c = 5 $$

Question 15

Which of the following is the correct factorization of $$ 27x^4$ - 8x $$?

Accepted Answer

$$ x(3x - 2)(9x^2$ + 6x + 4) $$

Question 16

Expand and simplify $$ (5 - 2x)^2$ .$$

Accepted Answer

$$ 25 - 20x + 4x^2$ $$

Question 17

A rectangle has sides $$ 2x + (y + 5) $$ and $$ 2x - (y + 5) . $$What is the area of the rectangle in simplified form?

Accepted Answer

$$ 4x^2$ - (y + 5)^2$ $$

Question 18

Factor completely: $$ 8r^2$ - 3rs + 10s^2$ .$$

Accepted Answer

$$ (4r + 5s)(2r - s) $$

Intermediate Algebra Exam 3 Review – Step-by-Step Guidance

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