Problem 75 - Elementary and Intermediate Algebra

Question

Multiply. See Example 8. $$ (x-4)(x+1)(x-3) $$

Step-by-Step Solution

Answer

The product is \(x^3 - 6x^2 + 5x + 12\).

Step 1: Multiply the First Two Binomials

We start by multiplying the first two binomials, \[(x-4)(x+1) = x(x+1) - 4(x+1) = x^2 + x - 4x - 4 = x^2 - 3x - 4.\] This simplifies the expression to \[(x^2 - 3x - 4)(x-3).\]

Step 2: Distribute Each Term of the Trinomial

Now, distribute each term of the trinomial \(x^2 - 3x - 4\) to the binomial \(x-3\). Start with \(x^2\): \[x^2(x-3) = x^3 - 3x^2.\]

Step 3: Continue Distribution with the Linear Term

Continue by distributing the \(-3x\) term to the binomial: \[-3x(x-3) = -3x^2 + 9x.\]

Step 4: Finish Distribution with the Constant Term

Finally, distribute the \(-4\) term to the binomial: \[-4(x-3) = -4x + 12.\]

Step 5: Combine Like Terms

Combine all the results from Steps 2, 3, and 4. Add and organize like terms for the polynomial: \[x^3 - 3x^2 - 3x^2 + 9x - 4x + 12.\] Combine like terms: \[x^3 - 6x^2 + 5x + 12.\]

Key Concepts

Understanding Binomials

A binomial is a type of polynomial that contains exactly two terms. These terms can be any combination of variables and numbers, but are often seen as two distinct pieces connected by either addition or subtraction.
Examples of binomials include expressions like \(x + 1\), \(x - 4\), or even \(3y^2 - 5\). The expressions you see here are short and simple, which makes them "bi-" (meaning two) "-nomial" expressions.
When you multiply two or more binomials, like in our original exercise \((x-4)(x+1)\), the goal is to expand and simplify them. This requires following steps to distribute each part of one binomial across each part of another binomial.

Exploring Trinomials

A trinomial goes one step further from a binomial by including three terms in the expression. These terms are again combined using addition or subtraction.
For instance, a trinomial looks something like \(x^2 - 3x - 4\), which is what you find when you initially multiply and simplify two binomials.
The process of working with trinomials often involves distributing each of its three terms across another expression, as we see when we multiply a trinomial by a binomial, such as in the steps that work with \((x^2 - 3x - 4)(x-3)\). It makes using distribution a bit more extended, but the principle remains consistent.

The Concept of Distribution

Distribution is a key technique used in algebra to simplify expressions. When you distribute, you take each term inside one set of parentheses and multiply it by each term inside another set.
This approach helps to eliminate the parentheses and expands the expressions into a longer polynomial. For instance, in step 2 of our solution, each term of the trinomial \(x^2 - 3x - 4\) is distributed over the binomial \(x-3\).

• This results in multiplying \(x^2\) with \(x\) and \(-3\).
• Then, multiply \(-3x\) with \(x\) and \(-3\).
• Lastly, \(-4\) with \(x\) and \(-3\).

Navigating through each of these distributions gives you the individual product terms that eventually come together for the complete polynomial.

Recognizing and Combining Like Terms

Finally, we come to combining like terms, which is crucial for simplifying the expression after distribution.
Like terms are terms that have the same variable raised to the same power. For example, \(x^2\) terms can be added or subtracted with other \(x^2\) terms, and \(x\) terms with other \(x\) terms.
In the final step of our exercise, you identify and combine these like terms:

• First, consolidate all \(x^2\) terms, such as \(-3x^2 - 3x^2\), which becomes \(-6x^2\).
• Then, handle the \(x\) terms as \(9x - 4x\), resulting in \(5x\).

The entire combination gives us a simplified polynomial \(x^3 - 6x^2 + 5x + 12\), clearly showcasing the power of recognizing and managing like terms.