Factor by Grouping a^2b^2+a
The problem asks for the application of the factorization technique known as "factor by grouping". It presents an algebraic expression, specifically "a^2b^2 + a", and the objective is to reorganize and rewrite this expression by grouping terms in a way that common factors can be identified and factored out, thereby simplifying the expression into a product of factors. Keep in mind that factor by grouping usually requires an expression with four terms to be most effective, so solving the problem may require creative manipulation of the expression or recognizing a different factoring strategy.
$a^{2} b^{2} + a$
Answer
Solution:
Step 1:
It is not possible to factor the given polynomial using the grouping method. Consider alternative factoring strategies or use the Factor command if uncertain.
Step 2:
Extract the common factor $a$ from the expression $a^{2}b^{2} + a$.
Step 2.1:
Remove the factor $a$ from $a^{2}b^{2}$ to get $a(ab^{2}) + a$.
Step 2.2:
Express $a$ as $a^{1}$ to maintain consistency in the expression $a(ab^{2}) + a^{1}$.
Step 2.3:
Take out the factor $a$ from $a^{1}$, resulting in $a(ab^{2}) + a \cdot 1$.
Step 2.4:
Factor out $a$ from the entire expression to obtain $a(ab^{2} + 1)$, simplifying to $a(ab^{2} + 1)$.
Knowledge Notes:
The process of factoring by grouping involves rearranging the terms of a polynomial and factoring out common factors from each pair of terms. However, not all polynomials can be factored by grouping. In this case, the polynomial $a^{2}b^{2} + a$ does not have a grouping that allows for this method to be used directly.
The alternative approach taken here is to factor out the greatest common factor (GCF), which is the largest expression that divides all terms in the polynomial. For the given polynomial, the GCF is $a$.
Here are the steps broken down:
Identify the GCF: Look at all terms in the polynomial and identify the highest power of any common variables or numbers that can be factored out.
Factor out the GCF: Rewrite each term of the polynomial as a product of the GCF and another term. This often simplifies the polynomial.
Rewrite the expression: After factoring out the GCF, rewrite the polynomial as a product of the GCF and a new polynomial that contains the remaining terms.
Check your work: Always verify that when you distribute the GCF back into the polynomial, you get the original expression.
In this solution, the process of factoring out $a$ from each term is shown step by step, ultimately leading to the simplified form of the polynomial.
