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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]
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