Problem 55 - 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. $$(x-4)\left(x^{2}+5 x-4\right)$$

Step-by-Step Solution

Answer

The product is \(x^3 + x^2 - 24x + 16\).

Step 1: Understand the Problem

We need to find the product of the binomial \((x-4)\) with the trinomial \((x^2 + 5x - 4)\). This type of problem involves distributing each term of the binomial across the trinomial.

Step 2: Distribute the First Term

Multiply the first term of the binomial, \(x\), with each term in the trinomial. \[ x imes x^2 = x^3 \ x imes 5x = 5x^2 \ x imes -4 = -4x \]So, the result is \(x^3 + 5x^2 - 4x\).

Step 3: Distribute the Second Term

Multiply the second term of the binomial, \(-4\), with each term in the trinomial. \[ -4 imes x^2 = -4x^2 \ -4 imes 5x = -20x \ -4 imes -4 = 16 \]So, the result is \(-4x^2 - 20x + 16\).

Step 4: Combine Like Terms

Add the results from Steps 2 and 3 together and combine like terms:\[x^3 + 5x^2 - 4x - 4x^2 - 20x + 16\]Combine terms:\[x^3 + (5x^2 - 4x^2) + (-4x - 20x) + 16 = x^3 + x^2 - 24x + 16\]

Step 5: Finalize the Answer

After combining like terms, the final product of given expression is \[x^3 + x^2 - 24x + 16\].

Key Concepts

Binomial

A binomial is a polynomial with exactly two terms. These two terms are connected by either a plus sign or a minus sign. For example, the expression \((x-4)\) is a binomial because it consists of two terms: \(x\) and \(-4\). Binomials are a fundamental building block in algebra, and knowing how to manipulate and multiply them is crucial. When multiplying them with other polynomials, like trinomials, we need to apply the distribution method.

Trinomial

A trinomial is a polynomial composed of three terms. These terms are also combined by plus or minus signs. In the expression \((x^2 + 5x - 4)\), we see that it is made up of three distinct terms: \(x^2\), \(5x\), and \(-4\). Trinomials appear often in algebra, especially in quadratic equations. They require special attention when distributing and combining like terms during algebraic operations.

Distribution Method

The distribution method is a technique used for multiplying polynomials, such as a binomial with a trinomial. It involves multiplying each term of one polynomial by each term of another. Consider multiplying \((x-4)\) by \((x^2 + 5x - 4)\). We first take the term \(x\) from the binomial and multiply it through each term in the trinomial to get \(x^3 + 5x^2 - 4x\). Next, we do the same with \(-4\), resulting in \(-4x^2 - 20x + 16\). Then, we must combine the results.

Like Terms

In algebra, 'like terms' are terms that have identical variables and powers, allowing them to be combined in expressions. This is an essential part of simplifying polynomials after performing operations like multiplication. In the expression \(x^3 + 5x^2 - 4x - 4x^2 - 20x + 16\), \(5x^2\) and \(-4x^2\) are like terms because they both involve \(x^2\). Similarly, \(-4x\) and \(-20x\) are like terms. Combining these allows us to simplify the expression to \(x^3 + x^2 - 24x + 16\). Recognizing and correctly combining like terms is crucial for achieving the simplest form of a polynomial.