factor completely.10xy 35x 6y 21enter your answer in the box.

Lussy04

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24-01-2025

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Factoring Expressions: A Step-by-Step Guide to Factoring 10xy + 35x + 6y + 21

Factoring algebraic expressions is a fundamental skill in algebra. It involves breaking down a complex expression into simpler, multiplied components. This process is crucial for solving equations, simplifying expressions, and understanding mathematical relationships. Let's explore how to completely factor the expression 10xy + 35x + 6y + 21.

Understanding Factoring Techniques

Before tackling our specific expression, let's review some common factoring methods:

• Greatest Common Factor (GCF): This is the simplest method. Identify the largest number and/or variable that divides all terms evenly. Factor this GCF out of the expression.

• Grouping: When an expression has four or more terms, grouping can be effective. Group related terms together and then factor each group separately. Look for common factors within each group. Often, after factoring the groups, you'll find a common factor remaining.

• Difference of Squares: This method applies to expressions in the form a² - b², which factors to (a + b)(a - b).

• Trinomial Factoring: This is used for expressions with three terms (trinomials), often in the form ax² + bx + c. The goal is to find two binomials that multiply to give the trinomial.

Factoring 10xy + 35x + 6y + 21

Our expression, 10xy + 35x + 6y + 21, has four terms. The most appropriate technique here is grouping.

Step 1: Group related terms.

We can group the terms as follows: (10xy + 35x) + (6y + 21)

Step 2: Factor each group.

• Group 1 (10xy + 35x): The greatest common factor is 5x. Factoring this out, we get 5x(2y + 7).

• Group 2 (6y + 21): The greatest common factor is 3. Factoring this out, we get 3(2y + 7).

Step 3: Look for common factors across groups.

Notice that both factored groups have the common factor (2y + 7).

Step 4: Factor out the common factor.

We can now factor out (2y + 7) from the entire expression: (2y + 7)(5x + 3)

Final Answer

Therefore, the completely factored form of 10xy + 35x + 6y + 21 is (2y + 7)(5x + 3). This represents the expression as a product of two simpler expressions. You can check your answer by expanding this factored form using the FOIL method (First, Outer, Inner, Last) to ensure you arrive back at the original expression.

Practice Problems

To solidify your understanding, try factoring these expressions using the techniques outlined above:

• 12ab + 8a - 9b - 6
• 4mn + 16m - 5n - 20
• 15pq + 20p - 21q - 28

By consistently practicing factoring techniques, you will improve your algebraic skills and become more confident in tackling more complex problems. Remember to always check your work by expanding your factored answer.