Question 1

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

Accepted Answer

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

Question 2

Given the polynomial $$2x^2 - 3x + 8x - 12$, what is the sum of its roots?

Accepted Answer

Question 3

If $$2x^2 - 3x + 8x - 12 = 0$, what are the solutions for x$?

Accepted Answer

$$ x = 3, \ x = -4 $$

Question 4

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

Accepted Answer

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

Question 5

A chemical reaction's rate is modeled by the quadratic function $$R(t) = 2t^2$ - 3t + 8t - 12. At $$what time $$t$$ does the rate become zero?

Accepted Answer

$$ t = 3  or$$ $$ t = -4 $$

Question 6

Which of the following is the correct first step in factoring $$2x^2 - 3x + 8x - 12$ by grouping?

Accepted Answer

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

Question 7

A company models its profit with $$P(x) = 2x^2$ - 3x + 8x - 12. $$For what values of $$x is $$the profit zero?

Accepted Answer

$$ x = 3 $$ and $$ x = -4 $$

Question 8

Which of the following is NOT a valid method for factoring $$2x^2 - 3x + 8x - 12$?

Accepted Answer

Factoring as a difference of squares

Question 9

If $$2x^2 - 3x + 8x - 12$ models the height (in meters) of a projectile at time x$ seconds, when does it hit the ground?

Accepted Answer

$$ x = 3  s $$and $$ x = -4  s$$

Question 10

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

Accepted Answer

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

Question 11

Suppose $$2x^2 - 3x + 8x - 12$ represents the net change in a population over time x$. If the population returns to its original value, what are the possible values of x$?

Accepted Answer

$$ x = 3 $$ and $$ x = -4 $$

Question 12

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

Accepted Answer

Factoring $$x$$ from $$2x^2 - 3x$ and 4$ from 8x - 12$

Question 13

If $$2x^2 - 3x + 8x - 12$ is set equal to zero, what is the discriminant of the resulting quadratic equation?

Accepted Answer

$$ 5^2$ - 4 \cdot 2 \cdot (-12) = 25 + 96 = 121 $$

Question 14

A business's revenue is modeled by $$R(x) = 2x^2$ - 3x + 8x - 12. $$What is the revenue when $$x = 2$$?

Accepted Answer

$$ 2(2)^2$ - 3(2) + 8(2) - 12 = 8 - 6 + 16 - 12 = 6 $$

Question 15

Which of the following is the correct quadratic equation after combining like terms in $$2x^2 - 3x + 8x - 12$?

Accepted Answer

$$ 2x^2$ + 5x - 12 $$

Question 16

If $$2x^2 - 3x + 8x - 12$ represents the area of a rectangular garden in square meters, and the width is x + 4$ meters, what is the length?

Accepted Answer

$$ 2x - 3 $$ meters

Factoring Polynomials by Grouping: Precalculus Study Notes

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