Problem 26 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(x-1)(x+4)(x-6)$$

Step-by-Step Solution

Answer

The simplified product is \(x^3 - 3x^2 - 22x + 24\).

Step 1: Recognize the Expression Type

The expression \((x-1)(x+4)(x-6)\) consists of three binomials. To simplify this expression, we can first multiply two binomials and then multiply the result by the remaining binomial.

Step 2: Multiply the First Two Binomials

Start by using the distributive property (also known as the FOIL method for binomials) on \((x-1)(x+4)\). This means:\[(x-1)(x+4) = x(x) + x(4) - 1(x) - 1(4) = x^2 + 4x - x - 4\]Combine like terms to get:\[x^2 + 3x - 4\].

Step 3: Multiply the Result with the Third Binomial

Now take the resulting quadratic expression \(x^2 + 3x - 4\) and multiply it by the third binomial \(x-6\):\[(x^2 + 3x - 4)(x-6)\]Use the distributive property across all terms:\[x^2(x-6) + 3x(x-6) - 4(x-6)\]Calculate each:\[x^3 - 6x^2 + 3x^2 - 18x - 4x + 24\]Combine like terms to simplify:\[x^3 - 3x^2 - 22x + 24\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions involving the multiplication of terms inside parentheses. It's a versatile tool that helps break down complex problems into more manageable parts. This property is often used to multiply a single term by each term inside a set of parentheses. Here’s the general form:\[ a(b + c) = ab + ac \]This means that you distribute the multiplication of \(a\) to both \(b\) and \(c\). By applying the distributive property, you ensure every term inside the parentheses is equally affected by the multiplication.

When multiplying binomials like in the expression \((x-1)(x+4)\), you'll use this property to distribute each term in one binomial across all terms in the other binomial. This step-by-step expansion significantly reduces the complexity of polynomial expressions.

FOIL Method

The FOIL Method is a specialized application of the distributive property and is a handy acronym for remembering the steps when multiplying two binomials. FOIL stands for First, Outer, Inner, and Last:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

For instance, to find the product of \((x-1)(x+4)\), apply the FOIL method:

• First: \(x \cdot x = x^2\)
• Outer: \(x \cdot 4 = 4x\)
• Inner: \(-1 \cdot x = -x\)
• Last: \(-1 \cdot 4 = -4\)

Add these products together: \(x^2 + 4x - x - 4\). Next, you'll move on to combining these like terms.

Combining Like Terms

Combining like terms is another essential procedure in simplifying algebraic expressions. After multiplying expressions using the distributive property or the FOIL method, you'll often end up with terms that can be combined. Like terms are terms that have the same variable raised to the same power. For example, \(3x\) and \(-x\) are like terms because they both contain \(x\) to the first power.

In the expression \(x^2 + 4x - x - 4\), you can see the terms \(4x\) and \(-x\). Since these terms are like terms, you can combine them by adding their coefficients:\[4x - x = 3x\]This results in a simpler expression: \(x^2 + 3x - 4\). Combining like terms streamlines your expression, making the solution more compact and easier to interpret. It's a critical step that ensures your expressions are as simplified as possible.