Problem 72 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 7 x-21+x^{3}-2 x^{2} $$

Step-by-Step Solution

Answer

Not factorable by grouping.

Step 1: Identify Terms to Group

We have the polynomial \( 7x - 21 + x^3 - 2x^2 \). Look for pairs of terms that can be grouped for factoring. Group the polynomial into two pairs: \( (x^3 - 2x^2) + (7x - 21) \).

Step 2: Factor Each Group Separately

First, factor out the greatest common factor (GCF) from each group.\( x^3 - 2x^2 \) can be factored as \( x^2(x - 2) \). The second group, \( 7x - 21 \), can be factored as \( 7(x - 3) \).

Step 3: Check for Common Binomial Factor

After factoring each group separately, check for a common binomial factor. Unfortunately, \( x^2(x - 2) \) and \( 7(x - 3) \) do not share a common binomial factor, making it impossible to factor the polynomial by grouping.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used for polynomials, especially when dealing with four-term polynomials.
This approach involves rearranging and combining terms into groups, which makes it easier to factor them separately.
Here's how you can leverage this method:

• Break the polynomial into pairs of terms.
• Factor out the greatest common factor (GCF) from each pair.
• If the results share a binomial factor, you can continue to factor the polynomial completely by extracting this binomial.

In our original exercise, after arranging the terms as \[(x^3 - 2x^2) + (7x - 21),\]we proceed to find the GCF for both pairs and factor them. Unfortunately, not all polynomials are factorable by this method, as observed in this situation, where a common binomial factor could not be found.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the highest number or expression that evenly divides two or more terms.
Finding the GCF is a crucial step in factoring polynomials because it simplifies each group, making polynomial manipulation easier.
To extract the GCF:

• Identify terms that share common variables or numbers.
• Factor out the highest power of each shared variable or number from both terms.

In the example \[x^3 - 2x^2,\]the GCF is \(x^2\), resulting in \[x^2(x - 2).\]With \[7x - 21,\]the GCF is 7, leading to \[7(x - 3).\]Properly identifying and extracting the GCF is essential to factor polynomials efficiently.

Binomial Factors

A binomial factor is a polynomial with two terms.
After factoring by grouping or finding the GCF within the groups, we often end up with binomial expressions that help further simplify the polynomial.
To identify binomial factors:

• Look at the factored forms from each group.
• Check if any binomial appears in both groups.

In our example, after factoring the groups as \[x^2(x - 2) + 7(x - 3),\]we see there is no common binomial factor like \[(x + a), (x - b),\] etc., shared between both expressions.When a common binomial factor is found, you can use it to combine the groups, leading to a further simplified or completely factored polynomial. However, in cases like this one, the absence of a common binomial factor means that factorization by grouping is not possible.