Circuit Training – Factoring by Grouping Beginning in Cell #1, factor the polynomial by the method of grouping. When you find a factor, continue in the same manner until you complete the circuit: Answer: a + x + 3 #1 2x^2 + 2ax + zx + 7a
The Correct Answer and Explanation is:
Correct Answer: (2x + z)(x + a)
Explanation
Factoring by grouping is a technique used on polynomials, typically those with four terms, when a single greatest common factor does not exist for all terms. The goal is to group the terms into pairs, factor the greatest common factor (GCF) from each pair, and then factor out a common binomial expression.
Step 1: Arrange and Group the Terms
The given polynomial is 2x² + 2ax + zx + za. The initial step is to group the four terms into two pairs. A successful grouping will create pairs that share a common factor, which in turn will lead to a common binomial factor for the entire expression. Let’s group the first two terms and the last two terms:
(2x² + 2ax) + (zx + za)
Step 2: Factor the Greatest Common Factor (GCF) from Each Group
Next, identify and factor out the GCF from each individual group.
For the first group, (2x² + 2ax), the GCF is 2x. Factoring this out leaves x + a. The expression becomes:
2x(x + a)
For the second group, (zx + za), the GCF is z. Factoring this out also leaves x + a. The expression becomes:
z(x + a)
Step 3: Combine the Factored Groups
Now, place the factored groups back into the main expression:
2x(x + a) + z(x + a)
Step 4: Factor Out the Common Binomial
Observe that both parts of the expression now share a common binomial factor, which is (x + a). This common binomial is the first factor of the final answer. To find the second factor, group the terms that were originally factored out from each pair (2x and +z).
The final factored form is obtained by combining the common binomial with the group of GCFs:
(x + a)(2x + z)
This is the fully factored form of the polynomial. For the circuit training, the factor (x + a) would be used to locate the next cell to solve.
