Problem 34 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Check your answer. $$10+7 k+k^{2}$$

Step-by-Step Solution

Answer

The short version of the answer is: \( (k+2)(k+5) \)

Step 1: Factor by Grouping

First, we will rewrite the quadratic expression as a four-term expression by regrouping the two constant terms \(7k\) and \(10\) as \(2k+5k+10\), where the first two terms have the greatest common factor (GCF) of \(k\), and the last two terms, the GCF is 5. So, the expression becomes: \[k^2 + 2k + 5k + 10\]

Step 2: Factor Each Group Separately

Next, we will factor out the GCF from each group separately: \[ k(k + 2) + 5(k + 2) \]

Step 3: Factor the Common Binomial

Now, we have a common binomial \((k + 2)\) in both terms, so we can factor that out: \[ (k + 2)(k + 5) \] So, the factored form of the given expression is \((k+2)(k+5)\).

Step 4: Check Your Answer

Now, we will check our answer by expanding the factored expression to see if it matches the original expression: \[ (k + 2)(k + 5)= k^2+5k+2k+10 = k^2 + 7k + 10 \] The factored form \((k+2)(k+5)\) correctly expands back into the original expression \(k^2 + 7k + 10\), so we have successfully factored the expression completely.

Key Concepts

Greatest Common Factor (GCF)

Finding the greatest common factor, or GCF, is a key step in factoring expressions. The GCF is the largest factor that can be divided evenly into all terms of a polynomial. When factoring, we first identify the GCF of different parts of the expression.

• In the given expression, the terms were regrouped such that the first two terms were combined and the last two terms were combined.
• For example, in the group \(k^2 + 2k\), the GCF is \(k\).
• In the group \(5k + 10\), the GCF is 5.

By finding these factors, we simplify each group before moving on to the next step in the factoring process.

Binomial Factorization

Binomial factorization involves looking for binomial expressions that can be factored out of a polynomial. Binomials are polynomial expressions that consist of exactly two terms.

• Once we have identified and factored out the GCF from individual groups, we often search for common binomial factors within the expression.
• In this case, after factoring each group separately, the expression turns into \(k(k + 2) + 5(k + 2)\).
• The common binomial factor here is \((k + 2)\), which appears in both groupings.

We then extract this common factor to simplify the whole expression further.

Quadratic Expression

A quadratic expression is a polynomial with an exponent of 2 as the highest power of a variable. It typically takes the general form of \(ax^2 + bx + c\).

• In our example exercise, \(k^2 + 7k + 10\) represents a quadratic expression.
• The structure consists of the squared term \(k^2\), a linear term \(7k\), and a constant term, 10.
• Quadratic expressions are often factored to find their roots or to simplify calculations.

Understanding the structure allows us to use various techniques, such as factoring by grouping, to break down and simplify these expressions.

Factoring by Grouping

Factoring by grouping is a technique where you group terms in a polynomial to make factoring easier. This method is particularly useful when you have more than three terms or when other methods don't work.

• Our problem begins by regrouping the middle terms \(7k\) into two parts, i.e., \(2k\) and \(5k\), producing \(k^2 + 2k + 5k + 10\).
• Next, we factor each group separately by extracting the GCF, resulting in an expression like \(k(k + 2) + 5(k + 2)\).
• Finally, we notice a common binomial \((k + 2)\) in both groups and factor it out to yield the final, neatly factored form: \((k + 2)(k + 5)\).

By organizing the polynomial into groups with common factors, we achieve a simpler form efficiently, showcasing the power of this method in simplifying complex expressions.