Question

name
7.6 practice with calcchat and calcview
12.15.25
in exercises 1–6, factor the polynomial. example 1

1. ( 3a^2 + 3a - 6 )
2. ( 8x^2 + 8x - 48 )
3. ( 4k^2 + 28k + 48 )
4. ( 6y^3 - 24y + 18 )
5. ( 78c^2 - 63b + 140 )
6. ( 9x^2 - 36x - 45 )

in exercises 7–14, factor the polynomial. examples 2 and 3

1. ( 3b^2 + 11b + 6 )
2. ( 8m^2 + 30m + 7 )
3. ( 6x^2 - 5x + 1 )
4. ( 10w^2 - 31w + 15 )
5. ( 3n^2 + 5n - 2 )
6. ( 4c^2 + 4c - 3 )
7. ( 8g^2 - 10g - 12 )
8. ( 18v^2 - 15v - 18 )

in exercises 15–20, factor the polynomial. example 4

1. ( -3c^2 + 11t - 6 )
2. ( -7y^2 - 25y - 12 )
3. ( -4c^2 + 19c + 5 )
4. ( -8h^2 - 13h + 6 )
5. ( -15w^2 - w + 28 )
6. ( -22d^2 + 29d - 9 )

error analysis in exercises 21 and 22, describe and correct the error in factoring the polynomial.
21.
( \begin{align} 2x^2 - 2x - 24 &= 2(x^2 - 2x - 24) \\ &= 2(x - 6)(x + 4) end{align} )
(marked with red x)
22.
( 6x^2 - 7x - 3 = (3x - 3)(2x + 1) )
(marked with red x)

Explanation:

Let's solve problem 21 from the error analysis section.

Step 1: Analyze the given factoring

The original polynomial is \(2x^2 - 2x - 24\). The first step taken was \(2x^2 - 2x - 24 = 2(x^2 - 2x - 24)\). But when factoring out the GCF (Greatest Common Factor), we need to divide each term by 2. Let's check the middle term: \(\frac{-2x}{2}=-x\), not \(-2x\). So the error is in the factoring out of the GCF.

Step 2: Correctly factor out the GCF

First, factor out 2 from each term: \(2x^2 - 2x - 24 = 2(x^2 - x - 12)\)

Step 3: Factor the quadratic inside the parentheses

We need to find two numbers that multiply to \(-12\) and add to \(-1\). Those numbers are \(-4\) and \(3\). So \(x^2 - x - 12=(x - 4)(x + 3)\)

Step 4: Combine the factors

Putting it all together, \(2x^2 - 2x - 24 = 2(x - 4)(x + 3)\)

Answer:

The error was in factoring out the GCF (the middle term was incorrectly divided by 2). The correct factoring is \(2x^2 - 2x - 24 = 2(x - 4)(x + 3)\)