Q: If a trinomial $$ax^2 + bx + c$ can be factored by grouping, what must be true about the product a$$ \times c$$?

There exist two integers whose product is $$a \times c$$ and whose sum is $$b$$

Q: Which step is necessary before grouping terms when factoring a trinomial?

Find two numbers that multiply to $$a \times c$$ and add to $$b$$

Question 1

Which of the following is the correct factorization of the trinomial $$4x^2 - 19x + 12$ using the grouping method?

Accepted Answer

$$ (4x - 3)(x - 4) $$

Question 2

When factoring $$4x^2 - 19x + 12$ by grouping, which pair of numbers should be used to split the middle term?

Accepted Answer

$$-3$$ and $$-16$$

Question 3

Given the polynomial $$4x^2 - 19x + 12$, what is the first step in the grouping method?

Accepted Answer

Split the middle term into two terms whose coefficients add to $$-19$$ and whose product is $$4 	imes 12$$

Question 4

After splitting the middle term of $$4x^2 - 19x + 12$ into -3x$ and -16x$, which is the correct grouping for further factoring?

Accepted Answer

$$4x^2 - 3x$ and -16x + 12$

Question 5

What is the greatest common factor (GCF) of the terms $$4x^2 - 3x$?

Accepted Answer

$$x$$

Question 6

After factoring $$x$$ from $$4x^2 - 3x$ and -4$ from -16x + 12$, what is the resulting expression?

Accepted Answer

$$x(4x - 3) - 4(4x - 3)$$

Question 7

Which of the following is a correct application of the grouping method to factor $$4x^2 - 19x + 12$?

Accepted Answer

Split $$-19x$$ into $$-3x$$ and $$-16x$$, group as $$(4x^2 - 3x) + (-16x + 12$$)$$, factor each group, then factor out the common binomial.

Question 8

If a trinomial $$ax^2 + bx + c$ can be factored by grouping, what must be true about the product a$$ 	imes c$$?

Accepted Answer

There exist two integers whose product is $$a 	imes c$$ and whose sum is $$b$$

Question 9

Which of the following is NOT a valid step in the grouping method for factoring trinomials?

Accepted Answer

Combine the first and last terms before splitting the middle term

Question 10

A chemical reaction produces a quadratic equation for the concentration of a product: $$4x^2 - 19x + 12 = 0$. What are the possible concentrations (values of x$) using factoring?

Accepted Answer

$$x = 4 or$$ $$x = 3$$

Question 11

If you factor $$4x^2 - 19x + 12$ as (4x - 3)(x - 4$$)$$, which values of x$ make the expression zero?

Accepted Answer

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

Question 12

A student attempts to factor $$4x^2 - 19x + 12$ and gets (2x - 3)(2x - 4$$)$$. What error did the student make?

Accepted Answer

Incorrectly split the middle term and factored each group incorrectly

Question 13

Which real-world scenario could be modeled by a trinomial like $$4x^2 - 19x + 12$?

Accepted Answer

The profit from selling $$x$$ units of a product, where costs and revenues are quadratic and linear

Question 14

If $$4x^2 - 19x + 12$ represents the height of a projectile at time x$, what does factoring the expression help you find?

Accepted Answer

The times when the projectile is at ground level

Question 15

Which of the following is the correct factorization of $$4x^2 - 19x + 12$?

Accepted Answer

$$(4x - 3)(x - 4)$$

Question 16

Which step is necessary before grouping terms when factoring a trinomial?

Accepted Answer

Find two numbers that multiply to $$a 	imes c$$ and add to $$b$$

Question 17

If you factor $$4x^2 - 19x + 12$ and get (4x - 3)(x - 4$$)$$, what is the sum of the roots?

Accepted Answer

$$4 + \frac{3}{4} = \frac{19}{4}$$

Question 18

Which of the following is a common mistake when using the grouping method to factor trinomials?

Accepted Answer

Choosing numbers that add to $$b$$ but do not multiply to $$a 	imes c$$

Factoring Trinomials Using the Grouping Method

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