Answered

At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

Determine if Ginny factored correctly. If not, explain where she made an error.

Ginny factored [tex]\(6x^2 - 31x - 30\)[/tex] as shown:
1. [tex]\(ac = -180\)[/tex] and [tex]\(b = 31\)[/tex]
2. [tex]\(36 \cdot (-5) = -180\)[/tex] and [tex]\(36 + (-5) = 31\)[/tex]
3. [tex]\(6x^2 + 36x - 5x - 30\)[/tex]
4. [tex]\(6x(x + 6) - 5(x + 6)\)[/tex]
5. [tex]\((x + 6)(6x - 5)\)[/tex]

Options:
A. Ginny made a mistake in step 1 when she identified [tex]\(b = 31\)[/tex]. It should be [tex]\(b = -31\)[/tex].
B. Ginny was correct until step 3 when she used 36 and -5 as the coefficients of [tex]\(x\)[/tex].
C. Ginny was correct until step 4 when she incorrectly factored the GCF from [tex]\(5x - 30\)[/tex].
D. Ginny factored the trinomial correctly.

Sagot :

GinnyAnswer
To determine whether Ginny factored the polynomial [tex]\(6x^2 - 31x - 30\)[/tex] correctly, let's analyze each step she took and identify any mistakes.

### Step-by-Step Analysis:
1. Given Polynomial:
[tex]\[6x^2 - 31x - 30\][/tex]

2. Step 1: Calculate [tex]\( ac \)[/tex] and Identify [tex]\( b \)[/tex]
[tex]\[ ac = 6 \times -30 = -180 \][/tex]
Ginny identifies [tex]\( b \)[/tex] incorrectly as [tex]\( 31 \)[/tex], but it should be [tex]\( -31 \)[/tex].

3. Step 2: Finding Factor Pairs
Ginny finds factor pairs [tex]\(36\)[/tex] and [tex]\(-5\)[/tex] for [tex]\(-180\)[/tex] and incorrectly because the sum [tex]\( 36 + (-5) = 31 \)[/tex]. However, we need the sum to be [tex]\(-31\)[/tex].

4. Step 3: Split the Middle Term with Incorrect Factors:
[tex]\[ 6x^2 + 36x - 5x - 30 \][/tex]
This expression is based on the incorrect factor pairs [tex]\(36\)[/tex] and [tex]\(-5\)[/tex].

5. Step 4: Grouping Terms:
[tex]\[ 6x(x + 6) - 5(x + 6) \][/tex]
This is Ginny’s further simplification from step 3, based on incorrect factor pairs.

6. Step 5: Factored Form:
[tex]\[ (x + 6)(6x - 5) \][/tex]
This is the final result after factoring.

### Correct Process:

1. Correct Polynomial Terms:
Given Polynomial:
[tex]\[ 6x^2 - 31x - 30 \][/tex]

2. Correct Multiplication of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[ ac = 6 \times -30 = -180 \][/tex]
Correct value of [tex]\( b \)[/tex] is [tex]\(-31\)[/tex].

3. Correct Factor Pairs for [tex]\(-180\)[/tex]:
We need pairs of numbers that multiply to [tex]\(-180\)[/tex] and sum to [tex]\(-31\)[/tex]. The correct pairs are [tex]\(-36\)[/tex] and [tex]\(5\)[/tex].

4. Correct Split Based on Factors:
[tex]\[ 6x^2 - 36x + 5x - 30 \][/tex]

5. Correct Grouping and Factoring Out the GCF:
[tex]\[ (6x^2 - 36x) + (5x - 30) \][/tex]
[tex]\[ 6x(x - 6) + 5(x - 6) \][/tex]

6. Correct Final Factored Form:
[tex]\[ (x - 6)(6x + 5) \][/tex]

### Conclusion:
Ginny made a mistake in multiple steps:

1. She identified [tex]\( b \)[/tex] as [tex]\( 31 \)[/tex] incorrectly. It should be [tex]\( -31 \)[/tex].
2. Ginny incorrectly used the factor pairs [tex]\(36\)[/tex] and [tex]\(-5\)[/tex]
3. Her factorization from step 3 onward is incorrect due to using these wrong factors.

Thus, the correct factored form of [tex]\(6x^2 - 31x - 30\)[/tex] is:
[tex]\[ (x - 6)(6x + 5) \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.