Question 6 (Essay Worth 4 points)
(Factoring Algebraic Expressions)

You are given the expression [tex]30x^4y^3 - 12x^3y[/tex].

Part A: Find a common factor for the expression that has a coefficient other than 1 and that contains at least one variable. (1 point)

Part B: Explain how you found the common factor. (1 point)

Part C: Rewrite the expression using the common factor you found in Part A. Show every step of your work. (2 points)



Answer :

### Part A: Finding a Common Factor

The given expression is [tex]\(30x^4y^3 - 12x^3y\)[/tex].

A common factor for this expression, containing at least one variable and having a coefficient other than 1, is [tex]\(6x^3y\)[/tex].

### Part B: Explanation of Finding the Common Factor

To determine the common factor:

1. Coefficients: First, we find the greatest common divisor (GCD) of the coefficients 30 and 12. The GCD of 30 and 12 is 6.

2. Variable [tex]\(x\)[/tex]: Next, we look at the powers of the variable [tex]\(x\)[/tex] in each term. The first term has [tex]\(x^4\)[/tex], and the second term has [tex]\(x^3\)[/tex]. The lowest power of [tex]\(x\)[/tex] common to both terms is [tex]\(x^3\)[/tex].

3. Variable [tex]\(y\)[/tex]: Finally, we consider the powers of the variable [tex]\(y\)[/tex] in each term. The first term has [tex]\(y^3\)[/tex], and the second term has [tex]\(y\)[/tex]. The lowest power of [tex]\(y\)[/tex] common to both terms is [tex]\(y\)[/tex].

Combining these results, we get the common factor [tex]\(6x^3y\)[/tex].

### Part C: Rewriting the Expression Using the Common Factor

To factor the expression [tex]\(30x^4y^3 - 12x^3y\)[/tex] using the common factor [tex]\(6x^3y\)[/tex]:

1. Extract the common factor:
[tex]\[ 30x^4y^3 - 12x^3y = 6x^3y \cdot \left( \frac{30x^4y^3}{6x^3y} \right) - 6x^3y \cdot \left( \frac{12x^3y}{6x^3y} \right) \][/tex]

2. Simplify each term inside the parentheses:
[tex]\[ = 6x^3y \cdot \left( 5xy^2 \right) - 6x^3y \cdot \left( 2 \right) \][/tex]

3. Combine the simplified terms:
[tex]\[ = 6x^3y \cdot (5xy^2 - 2) \][/tex]

Thus, the expression [tex]\(30x^4y^3 - 12x^3y\)[/tex] can be factored as:
[tex]\[ 30x^4y^3 - 12x^3y = 6x^3y(5xy^2 - 2) \][/tex]

In conclusion, the common factor of the given expression is [tex]\(6x^3y\)[/tex], and the factored form of the expression is [tex]\(6x^3y(5xy^2 - 2)\)[/tex].