Problem 25 - 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. $$(x+1)(x-2)(x-3)$$

Step-by-Step Solution

Answer

The product is \(x^3 - 4x^2 + x + 6\).

Step 1: Multiply the First Two Binomials

Start with multiplying the first two binomials, \( (x + 1)(x - 2) \), using the distributive property (or FOIL method for binomials). First, multiply \(x \) by \(x \) to get \((x^2)\). Then multiply \(x \) by \(-2\) to get \(-2x\). Next, multiply \(1 \) by \(x \) to get \(+x\). Finally, multiply \(1 \) with \(-2\) to get \(-2\). Combine like terms. This results in: \[x^2 - 2x + x - 2 = x^2 - x - 2.\]

Step 2: Multiply the Result with the Third Binomial

Now, take the result from Step 1, \(x^2 - x - 2\), and multiply it by the third binomial \(x - 3\). Use the distributive property again:- Multiply \(x^2 \) by \(x\) to get \(x^3\) and by \(-3\) to get \(-3x^2\).- Multiply \(-x\) by \(x\) to get \(-x^2\) and by \(-3\) to get \(3x\).- Multiply \(-2\) by \(x\) to get \(-2x\).- Multiply \(-2\) by \(-3\) to get \(6\).Combine like terms from these products, which results in:\[x^3 - 3x^2 - x^2 + 3x - 2x + 6 = x^3 - 4x^2 + x + 6.\]

Step 3: Simplify the Expression

Combine all like terms in the expression \(x^3 - 4x^2 + x + 6\). Notice there are no more like terms to combine. This is the final simplified form of the product.

Key Concepts

Binomial Multiplication

Binomial multiplication is a fundamental concept in algebra that involves multiplying two expressions, each containing two terms, known as binomials. For example, a binomial can take the form of

• (x + 1)
• (x - 2)

When multiplying binomials, the aim is to apply specific rules or methods to simplify the product into a polynomial with multiple terms.

Binomial multiplication is often a precursor to more complex operations in algebra. It lays the groundwork for understanding how variables and coefficients interact when combined through multiplication.

Distributive Property

The distributive property is a key concept in mathematics that facilitates multiplication over addition or subtraction within parentheses. Essentially, it states that for any three terms, a, b, and c, the equation \[a(b + c) = ab + ac\]holds true.

This property is particularly useful in binomial multiplication, as it allows you to distribute each term in one binomial across the terms in another. For instance, using the distributive property in multiplying \((x + 1)(x - 2)\),you multiply each term in the first binomial by each term in the second, systematically applying the property to simplify the expression.

The distributive property helps in breaking down complex expressions into smaller, easy-to-handle components, making it easier to manage and solve.

FOIL Method

The FOIL method is a mnemonic that helps remember the steps for multiplying two binomials quickly and accurately. FOIL stands for:

• First
• Outer
• Inner
• Last

These words represent the order in which terms should be multiplied when using this approach.

For the binomials \((x + 1)(x - 2)\),the FOIL method would proceed as follows:

• First: Multiply the first terms from each binomial: \(x \cdot x = x^2\)
• Outer: Multiply the outer terms: \(x \cdot (-2) = -2x\)
• Inner: Multiply the inner terms: \(1 \cdot x = x\)
• Last: Multiply the last terms: \(1 \cdot (-2) = -2\)

The FOIL method can be quite advantageous for students, as it offers a structured plan to ensure all parts of the multiplication are covered, helping to avoid missing any terms in the solution.

Combining Like Terms

Combining like terms is the process of simplifying an algebraic expression by merging terms that have identical variables raised to the same power. This is crucial in the process of polynomial multiplication to achieve the most simplified result.

During any polynomial operation, you might end up with several terms that are similar, like \(-2x + x\).Here, these are like terms because both have the variable x to the same degree. Simply adding their coefficients results in a combined term.

For example, in the expression \(x^2 - 2x + x - 2\),combining the like terms \(-2x\) and \(+ x \)yields a simpler expression: \(x^2 - x - 2\).
This step reduces the complexity and number of terms in an equation, making it more manageable. Always remember to check for and combine like terms during the simplification to ensure your solution is accurate and sleek.