Problem 70 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ p^{2}-8 p+7 p-56 $$

Step-by-Step Solution

Answer

The factored form is \( (p-8)(p+7). \)

Step 1 - Group the terms

First, group the terms in pairs: \( (p^{2}-8p)+ (7p-56) \)

Step 2 - Factor out the greatest common factor (GCF) from each group

Next, factor out the GCF from each pair: \( p(p-8)+7(p-8) \)

Step 3 - Factor by grouping

Now, factor out the common binomial factor \( (p-8)\): \( (p-8)(p+7) \)

Key Concepts

Greatest Common Factor

When factoring by grouping, identifying the greatest common factor (GCF) is important.
The GCF is the largest factor that divides each term in a set of terms.
For example, in the polynomial \(p^{2} - 8p + 7p - 56\), we group terms into pairs: \((p^{2} - 8p) + (7p - 56)\).
For the first group \(p^{2} - 8p\), the GCF is \(p\).
For the second group \(7p - 56\), the GCF is \(7\).
By factoring out these GCFs, we get \(p(p-8) + 7(p-8)\).

Binomial Factor

In algebra, a binomial is a polynomial with two terms.
Here, we notice a common binomial factor \((p-8)\).
Once we factor out the GCF from each group, our expression becomes \(p(p-8) + 7(p-8)\).
This form has a common binomial factor: \((p-8)\).
By factoring out the common binomial factor, we simplify the expression to \((p-8)(p+7)\).
This shows how the expression can be condensed neatly by focusing on common binomial factors.

Grouping Terms

Grouping terms is an initial step in factoring by grouping.
It helps in simplifying complex polynomials.
First, you pair the terms in a way that allows factoring out common factors easily.
In our exercise, we took \(p^{2} - 8p + 7p - 56\) and grouped them as \((p^{2} - 8p) + (7p - 56)\).
This grouping helps highlight common factors within each set of parentheses.
It results in simpler expressions which can then be factored down further.

Algebra

Factoring by grouping is an important technique in algebra for simplifying polynomials.
Algebra provides a structured way of solving mathematical problems, including the use of polynomials and factoring techniques.
In this exercise, we used algebraic methods to simplify a four-term polynomial by grouping and factoring.
This method leverages the distributive property \(a(b + c) = ab + ac\).
This allows us to factor the given polynomial expression and write it in a more simplified form, making it easier to handle in further algebraic manipulations.