Question 1

Which of the following is the correct factorization of the polynomial $$2x^2 - 3x + 8x - 12$ by grouping?

Accepted Answer

$$2x^2 - 3x + 8x - 12$$ = $$(2x^2 - 3x) + (8x - 12) = x(2x - 3) + 4(2x - 3) = (2x - 3)(x + 4$$)$$

Question 2

After factoring $$2x^2 - 3x + 8x - 12$ by grouping, what are the roots of the equation 2x^2$ - 3x + 8x - 12 = 0$$?

Accepted Answer

$$x = -4$$ and $$x = \frac{3}{2}$$

Question 3

Which of the following statements is true about the polynomial $$2x^2 - 3x + 8x - 12$?

Accepted Answer

It can be factored by grouping into two binomials.

Question 4

Suppose you are designing a rectangular garden with an area represented by $$2x^2 - 3x + 8x - 12$ square meters. If the length and width are the factors of this polynomial, what are their expressions?

Accepted Answer

Length: $$(2x - 3)$$, Width: $$(x + 4)$$

Question 5

If $$2x^2 - 3x + 8x - 12$ models the profit (in dollars) from selling x$ units of a product, for which values of x$ does the profit become zero?

Accepted Answer

$$x = -4$$ and $$x = \frac{3}{2}$$

Question 6

Which step is necessary to factor $$2x^2 - 3x + 8x - 12$ by grouping?

Accepted Answer

Group the terms as $$(2x^2 - 3x) + (8x - 12$$)$$

Question 7

After grouping $$2x^2 - 3x + 8x - 12$ as (2x^2$ - 3x) + (8x - 12)$$, what is the next step in the factorization process?

Accepted Answer

Factor $$x$$ from the first group and $$4$$ from the second group.

Question 8

Which of the following is a common mistake when factoring $$2x^2 - 3x + 8x - 12$ by grouping?

Accepted Answer

Failing to rearrange the terms to group like terms properly.

Question 9

If you set $$2x^2 - 3x + 8x - 12 = 0$ and solve for x$, which method is most efficient?

Accepted Answer

Factoring by grouping and then setting each factor to zero.

Question 10

A student claims that $$2x^2 - 3x + 8x - 12$ is a prime polynomial. Which of the following best explains why this is incorrect?

Accepted Answer

The polynomial can be factored by grouping into $$(2x - 3)(x + 4).$$

Question 11

If $$2x^2 - 3x + 8x - 12$ represents the area of a rectangle, and you know one side is x + 4$, what is the length of the other side?

Accepted Answer

$$2x - 3$$

Question 12

Which of the following is the expanded form of $$(2x - 3)(x + 4)$$?

Accepted Answer

$$2x^2 + 8x - 3x - 12$

Question 13

If $$2x^2 - 3x + 8x - 12$ models the revenue from ticket sales, and you want to find the break-even points, what should you do?

Accepted Answer

Set the polynomial equal to zero and solve for $$x.$$

Question 14

Which of the following is NOT a valid step in factoring $$2x^2 - 3x + 8x - 12$ by grouping?

Accepted Answer

Combining $$-3x$$ and $$8x$$ before grouping.

Question 15

If you factor $$2x^2 - 3x + 8x - 12$ and set it equal to zero, which equation do you solve to find the roots?

Accepted Answer

$$(2x - 3)(x + 4) = 0$$

Question 16

A company’s profit is modeled by $$2x^2 - 3x + 8x - 12$. If the company wants to know for which number of units sold the profit is zero, what should they do?

Accepted Answer

Factor the polynomial and solve for $$x$$ when the product equals zero.

Factoring Polynomials by Grouping

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