Question 1

A chemist is combining two solutions. The concentration of the first solution is represented by the polynomial $$3x^2 + 2x + 5$, and the second by 2x^2$ - x + 4. $$What is the resulting concentration polynomial after addition?

Accepted Answer

$$5x^2 + x + 9$

Question 2

A company’s profit in thousands of dollars is modeled by $$P(x) = 4x^2$ + 7x - 3. If $$their expenses are modeled by $$E(x) = 2x^2$ + 5x + 1$$, what is the polynomial representing their net profit $$P(x) - E(x)$$?

Accepted Answer

$$2x^2 + 2x - 4$

Question 3

Multiply the monomial $$-3x by $$the polynomial $$2x^2 - 5x + 4$. What is the resulting expression?

Accepted Answer

$$-6x^3$$ + $$15x^2 - 12x$

Question 4

A rectangular garden has a length of $$(x + 3)$$ meters and a width of $$(x - 2)$$ meters. What is the area of the garden in terms of $$x$$?

Accepted Answer

$$x^2 + x - 6$

Question 5

Find the square of the binomial $$(2x - 5). $$Which of the following is correct?

Accepted Answer

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

Question 6

Divide the polynomial $$12x^3$$ - $$6x^2 + 18x$ by the monomial 6x$. What is the result?

Accepted Answer

$$2x^2 - x + 3$

Question 7

Factor the trinomial $$x^2 + 7x + 12$.

Accepted Answer

$$(x + 3)(x + 4)$$

Question 8

Which of the following is a perfect square trinomial, and what is its factorization?

Accepted Answer

$$x^2 - 10x + 25$, which factors as (x$$ - $$5)^2$$

Question 9

Multiply the rational expressions $$\displaystyle \frac{x}{x+2} 	imes \frac{2}{x}$$ and simplify.

Accepted Answer

$$\displaystyle \frac{2}{x+2}$$

Question 10

Divide the rational expressions $$\displaystyle \frac{3x}{x-1} \div \frac{6}{x-1}$$ and simplify.

Accepted Answer

$$\displaystyle \frac{x}{2}$$

Question 11

Add the rational expressions $$\displaystyle \frac{5x}{x^2$ + 1} + \frac{2}{x^2$ + 1}$$ and simplify.

Accepted Answer

$$\displaystyle \frac{5x + 2}{x^2$ + 1}$$

Question 12

Subtract the rational expressions $$\displaystyle \frac{4x}{x^2$ - 9} - \frac{2}{x^2$ - 9}$$ and simplify.

Accepted Answer

$$\displaystyle \frac{4x - 2}{x^2$ - 9}$$

Question 13

Simplify $$\sqrt{50}$$ using the product rule for square roots.

Accepted Answer

$$5\sqrt{2}$$

Question 14

A rectangular field has an area of $$x^2 + 10x + 25$ square meters. What are the possible dimensions of the field?

Accepted Answer

$$(x + 5)$$ meters by $$(x + 5)$$ meters

Question 15

A scientist needs to dilute a solution. The concentration is modeled by $$x^2 - 4x + 4$. Factor the expression to find the concentration in terms of a binomial squared.

Accepted Answer

$$(x - 2)^2$

Question 16

Multiply the polynomials $$(x + 2)$$ and $$(x^2 - 3x + 4$$)$$. What is the result?

Accepted Answer

Question 17

A rectangular box has a volume given by $$6x^3$$ - $$9x^2 + 12x$. If the height is 3x$, what is the area of the base?

Accepted Answer

$$2x^2 - 3x + 4$

Question 18

Simplify $$\sqrt{72}$$ using the product rule for square roots.

Accepted Answer

$$6\sqrt{2}$$

Operations with Polynomials and Rational Expressions

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