Answered

Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.