Factor by Grouping 10a-25+5b

The given algebraic expression is "10a - 25 + 5b." The question asks to factor the expression by using the method of grouping. Factoring by grouping typically involves rearranging the terms (when possible) and grouping them in such a way that common factors can be factored out, simplifying the expression into a product of simpler expressions or factors.

$10 a - 25 + 5 b$

Answer

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Solution:

Step:1

The given expression cannot be simplified through the method of grouping. Consider an alternative strategy or opt for the Factor method if uncertain.

Step:2

Extract the common factor of $5$ from the expression $10 a - 25 + 5 b$ .

Step:2.1

Remove the factor of $5$ from $10 a$ . This results in $5 (2 a) - 25 + 5 b$ .

Step:2.2

Extract the factor of $5$ from $- 25$ . The expression now reads $5 (2 a) + 5 (- 5) + 5 b$ .

Step:2.3

Factor out $5$ from the terms $5 (2 a) + 5 (- 5)$ . The expression simplifies to $5 (2 a - 5) + 5 b$ .

Step:2.4

Finally, factor out $5$ from the entire expression $5 (2 a - 5) + 5 b$ . The fully factored form is $5 (2 a - 5 + b)$ , or simply $5 (2 a - 5 + b)$ .

Knowledge Notes:

When factoring by grouping, the goal is to rearrange and group the terms in such a way that they share a common factor. However, not all polynomials can be factored using this method. If the polynomial does not have a suitable arrangement of terms or does not have any common factors, then factoring by grouping is not possible.

In the given problem, the polynomial $10 a - 25 + 5 b$ does not have terms that can be grouped to factor by grouping. Instead, we look for a common factor across all terms. Here, the number $5$ is a common factor of all three terms. Factoring out the common factor simplifies the expression.

The process of factoring out the common factor involves dividing each term by the common factor and using the distributive property in reverse. The distributive property states that for any numbers $a$ , $b$ , and $c$ , the following is true: $a (b + c) = a b + a c$ . When factoring, we use this property to combine terms under a common factor.

In LaTeX, the multiplication symbol is denoted by \cdot or \times, and the factored expression is enclosed in parentheses. For example, factoring $5$ out of $10 a$ is written as $5 (2 a)$ in LaTeX.