Problem 73 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$(a+b) k^{2}+7(a+b) k-18(a+b)$$

Step-by-Step Solution

Answer

The completely factored expression of \((a+b)k^2 + 7(a+b)k - 18(a+b)\) is \((a+b)(k+9)(k-2)\).

Step 1: Identify the Greatest Common Factor (GCF)

The expression that we need to factor is: \[(a+b)k^2 + 7(a+b)k - 18(a+b)\] To find the GCF, we can see that each term has a common factor of \((a+b)\).

Step 2: Factor out the GCF

Now, we factor out the GCF \((a+b)\) from the expression: \[(a+b)(k^2 + 7k -18)\]

Step 3: Factor the Quadratic Expression inside the Parentheses

We will now focus on the quadratic expression inside the parentheses, which is: \[k^2 + 7k - 18\] Since it is a quadratic expression, we will try to factor it into the form of \((k + m)(k + n)\), where m and n are two numbers that multiply to -18 and add up to 7.

Step 4: Find the two numbers that satisfy the conditions

To find the two numbers that multiply to -18 and sum to 7, we can list down the factors of -18 and find the pair that adds up to 7: -18 x 1 = -18 18 x -1 = -18 -9 x 2 = -18 9 x -2 = -18 The pair that adds up to 7 is 9 and -2. So, m = 9 and n = -2.

Step 5: Factor the Quadratic Expression using m and n

Now that we have found the values of m and n, we can factor the quadratic expression inside the parentheses as follows: \[k^2 + 7k - 18 = (k + 9)(k - 2)\]

Step 6: Write down the final factored expression

So, combining the GCF and the factored quadratic expression inside the parentheses, we get the final factored expression: \[(a+b)k^2 + 7(a+b)k - 18(a+b) = (a+b)(k+9)(k-2)\] Therefore, the expression is completely factored as \((a+b)(k+9)(k-2)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a fundamental concept in factoring polynomials. When you factor using the GCF, you look for the largest expression that divides all terms in a polynomial without a remainder. This step simplifies the problem and makes further factoring easier.

In our exercise,

• each term had a common factor of \((a+b)\).
• Factoring out \((a+b)\) reduces the expression and exposes the core quadratic equation to be worked on next.

Identifying the GCF is usually the first step in the factoring process. It checks if the problem can be simplified at the outset.

Quadratic Expressions

A quadratic expression is a polynomial of degree 2, typically in the form \(ax^2 + bx + c\). Understanding its structure is crucial.

Quadratic expressions often need to be factored or solved, depending on the problem at hand. The standard format helps in recognizing opportunities to factor them into products of linear expressions.

In the problem given, after factoring out the GCF, we needed to factor \(k^2 + 7k - 18\). This expression fits the typical quadratic format, allowing us to apply standard factoring techniques. Recognizing these patterns is key to simplifying and solving polynomials.

Factoring Trinomials

Factoring trinomials is a specific skill under the umbrella of factoring quadratic expressions. It involves transforming expressions like \(k^2 + 7k - 18\) into the product of two binomials.

For successful factoring:

• Identify two numbers \(m\) and \(n\) that multiply to the constant term (here, \(-18\)) and add to the linear coefficient (here, \(7\)).
• The pair found was \(9\) and \(-2\), such that \(9 \times -2 = -18\) and \(9 + -2 = 7\).

Once these numbers are identified, you rewrite the quadratic as \((k + 9)(k - 2)\). This factorization simplifies the expression and is essential for finding roots or solutions. Understanding this process applies to many algebraic problems and is a powerful algebraic tool.