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An expression is shown below:

[tex]\[ 10n^3 - 15n^2 + 20xn^2 - 30xn \][/tex]

Part A: Rewrite the expression by factoring out the greatest common factor. (4 points)

Part B: Factor the entire expression completely. Show the steps of your work. (6 points)

Sagot :

GinnyAnswer
Certainly! Let's analyze and factor the given expression step by step.

Given expression:
[tex]\[ 10n^3 - 15n^2 + 20xn^2 - 30xn \][/tex]

### Part A: Rewrite the expression by factoring out the greatest common factor (GCF).

Step 1: Identify the constants/coefficients and variables involved.
- Coefficients: [tex]\(10, -15, 20, -30\)[/tex]
- Variables: [tex]\(n^3, n^2, xn^2, xn\)[/tex]

Step 2: Find the GCF of the coefficients.
- The GCF of [tex]\(10, 15, 20,\)[/tex] and [tex]\(30\)[/tex] is [tex]\(5\)[/tex].

Step 3: Find the GCF of the variable terms.
- All terms have at least one [tex]\(n\)[/tex]. The lowest power of [tex]\(n\)[/tex] among the terms is [tex]\(n\)[/tex].

Combining these, the GCF of the entire expression is [tex]\(5n\)[/tex].

Step 4: Factor out the GCF from each term.
[tex]\[ 10n^3 - 15n^2 + 20xn^2 - 30xn = 5n(2n^2 - 3n + 4xn - 6x) \][/tex]

Thus, the factored expression by removing the GCF is:
[tex]\[ 5n(2n^2 - 3n + 4xn - 6x) \][/tex]

### Part B: Factor the entire expression completely.

Step 5: Simplify inside the parentheses.

We need to check if we can factor further inside [tex]\(2n^2 - 3n + 4xn - 6x\)[/tex].

Rearrange the terms within the parentheses to group them for factoring:
[tex]\[ 2n^2 - 3n + 4xn - 6x = 2n^2 + 4xn - 3n - 6x \][/tex]

Step 6: Factor by grouping.

Group the terms in pairs:
[tex]\[ 2n^2 + 4xn - 3n - 6x = (2n^2 + 4xn) + (-3n - 6x) \][/tex]

Factor out common factors in each group:
[tex]\[ (2n^2 + 4xn) + (-3n - 6x) = 2n(n + 2x) - 3(n + 2x) \][/tex]

Step 7: Notice [tex]\(n + 2x\)[/tex] is a common factor in the groups.
[tex]\[ 2n(n + 2x) - 3(n + 2x) = (2n - 3)(n + 2x) \][/tex]

Therefore, the fully factored expression inside the parentheses is:
[tex]\[ 2n^2 - 3n + 4xn - 6x = (2n - 3)(n + 2x) \][/tex]

Step 8: Substitute back into the original factored expression.

Thus, the completely factored form of the original expression [tex]\(10n^3 - 15n^2 + 20xn^2 - 30xn\)[/tex] is:
[tex]\[ 5n(2n - 3)(n + 2x) \][/tex]

### Final Answer:

Part A:
[tex]\[ 5n(2n^2 - 3n + 4xn - 6x) \][/tex]

Part B:
[tex]\[ 5n(2n - 3)(n + 2x) \][/tex]
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