Problem 67 - Elementary Algebra

Question

For exercises 67-82, factor by grouping. Do not combine like terms before factoring. $$ x^{2}+7 x+4 x+28 $$

Step-by-Step Solution

Answer

(x + 7)(x + 4)

Step 1 - Group the terms

Group the terms into two pairs: o x² + 7x o 4x + 28

Step 2 - Factor out the greatest common factor from each pair

From the first group (x² + 7x), factor out x: x(x + 7) From the second group (4x + 28), factor out 4: 4(x + 7)

Step 3 - Factor out the common binomial factor

Notice that (x + 7) is a common factor: x(x + 7) + 4(x + 7) Factor out (x + 7): (x + 7)(x + 4)

Key Concepts

greatest common factor

The greatest common factor (GCF) is a crucial concept in factoring by grouping. It refers to the largest number or mathematical expression that divides all the terms in a group.
When factoring a polynomial like \(x^2 + 7x + 4x + 28\), we look for the GCF in each pair of terms.
In our example, we group the polynomial into two pairs: \(x^2 + 7x\) and \(4x + 28\).
We then identify the GCF for each pair:

• For \(x^2 + 7x\), the GCF is \(x\) because both terms have an \(x\) in common.
• For \(4x + 28\), the GCF is \(4\) because both terms are divisible by 4.

Factoring out the GCF simplifies the expression as it pulls out the common factors and helps in forming a simpler binomial factor.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into simpler 'factor' expressions that, when multiplied together, give back the original polynomial.
With the polynomial \(x^2 + 7x + 4x + 28\), we use grouping and factoring to simplify it.
First, we formed groups: \(x^2 + 7x\) and \(4x + 28\).
After factoring out the GCF from each group, we got:

• \(x(x + 7)\) from \(x^2 + 7x\)
• \(4(x + 7)\) from \(4x + 28\)

Combining these steps helps to express the original polynomial in a product of simpler polynomials, namely \((x + 7)(x + 4)\).
This process makes solving equations or simplifying expressions much easier.

common binomial factor

The common binomial factor is the binomial expression that appears in multiple terms and can be factored out to simplify an expression.
In our problem, after factoring the GCF from each pair, we get: \(x(x + 7) + 4(x + 7)\).
Here, \((x + 7)\) is a common factor in both expressions.
To factor out this common binomial factor, we treat \((x + 7)\) as we would a single variable or a number.
We can rewrite the expression as: \((x + 7)(x + 4)\).
Factoring out common binomial factors simplifies the polynomial, making it easier to work with and solve.