Problem 68 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ x^{2}+9 x+4 x+36 $$

Step-by-Step Solution

Answer

(x + 9)(x + 4)

Step 1 - Group the terms

First, group the terms in pairs as follows: ( x^2 + 9x ) ( + ) ( 4x + 36 ).

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

Factor out the GCF from each pair of terms: x^2 + 9x = x(x + 9) and 4x + 36 = 4(x + 9)

Step 3 - Factor out the common binomial factor

The expression now is: -x(x + 9) + 4(x + 9)- Factor out the common binomial factor ( x + 9 ) from both terms: ( x + 9 ) ( x + 4 )

Key Concepts

greatest common factor

The greatest common factor (GCF) is the highest number that divides exactly into two or more numbers. When dealing with algebraic expressions, the GCF is the largest expression that can divide each term in the polynomial without leaving a remainder. To find the GCF among the terms, you can:

• Identify the smallest power of each common variable in the terms.
• Look for the highest number that is a factor of all the coefficients.

For instance, if you have the terms \(x^2 + 9x\) and \(4x + 36\), the GCF for the first group is \(x\) because it appears in both terms. For the second group, the GCF is \(4\), since 4 is the largest number that can divide both 4 and 36.

algebraic expressions

An algebraic expression is a mathematical phrase that includes variables, numbers, and operation symbols. These expressions can represent real-world situations and can be simplified or manipulated to find values of variables.
For example, in the given problem \(x^2 + 9x + 4x + 36\), we see several terms:

• \(x^2\) - a term with a variable squared
• \(9x\) - a term with a variable multiplied by a coefficient
• \(4x\) - another term similar to \(9x\)
• \(36\) - a constant term

Algebraic expressions can be factored by grouping, simplifying, or other algebraic methods to solve for variables or simplify equations. Grouping is used here to rearrange and factor them more easily.

binomial factor

A binomial is an algebraic expression that contains exactly two terms. Factoring by grouping often leads to identifying a common binomial factor, which can then be factored out. In our example:
After factoring the GCF from each group:

• \(x^2 + 9x = x(x + 9)\)
• \(4x + 36 = 4(x + 9)\)

we see that \((x + 9)\) is a common binomial factor.
Thus, our expression becomes: \[x(x + 9) + 4(x + 9)\].
We can then factor \((x + 9)\) out: \[(x + 9)(x + 4)\]. The ability to recognize and factor binomials is crucial in simplifying and solving complex algebraic expressions.