Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((x+3)(x+4)\)

Step-by-Step Solution

Answer

Therefore, the product of \( (x+3) \) and \( (x+4) \) is \( x^2 + 7x + 12 \).

Step 1: Use the FOIL method

Apply the FOIL method: Multiply the First terms together (\(x * x = x^2\)), then the Outer terms (\(x * 4 = 4x\)), followed by the Inner terms (\(3 * x = 3x\)), and the Last terms (\(3 * 4 = 12\)).

Step 2: Combine like terms

Combine the like terms from the result of the First, Outer, Inner, Last multiplication. So the sum of \(4x\) and \(3x\) is \(7x\). Thus, the entire expression should now read as \(x^2 + 7x + 12\).

Key Concepts

FOIL Method

The FOIL method is a popular procedure used to multiply two binomials. It's named for the order in which you combine terms: First, Outer, Inner, Last. This structured approach helps ensure you don't miss any steps in multiplication, making it easier to simplify algebraic expressions efficiently.

When you multiply two binomials like

• First: Multiply the first terms in each binomial. For $\backslash ((x+3)(x+4)\backslash )$, this is $\backslash (x\; \backslash cdot\; x\; =\; x^2\backslash )$.
• Outer: Multiply the outermost terms from the binomials. Here, it’s $\backslash (x\; \backslash cdot\; 4\; =\; 4x\backslash )$.
• Inner: Multiply the innermost terms. This involves $\backslash (3\; \backslash cdot\; x\; =\; 3x\backslash )$.
• Last: Multiply the last terms in each binomial. Lastly, $\backslash (3\; \backslash cdot\; 4\; =\; 12\backslash )$.

The FOIL method breaks down the calculation into simple, manageable parts. This guidance ensures you systematically approach and expand every binomial multiplication.

Polynomial Multiplication

Polynomial multiplication involves taking two polynomial expressions and multiplying them to form a new polynomial. It’s important to distribute each term of the first polynomial by each term of the second, ensuring no terms are left out.

In our example with $\$(x+3)(x+4)\$$, by applying the FOIL method:

• We start with multiplying the first terms to get $\$x^2\$$.
• Then, the outer and inner multiplications yield $\$4x\$$ and $\$3x\$$ respectively.
• Finally, multiplying the last terms gives us $\$12\$$.

The process ensures that every term from each binomial affects the final polynomial. This systematic approach is essential, ensuring that all terms are accounted for.

Combining Like Terms

When dealing with polynomials, combining like terms is a vital skill in simplifying expressions. Like terms are terms that have the same variables raised to the same power. By combining them, you can simplify complex expressions into more manageable ones.

In the expression derived from $\$(x+3)(x+4)\$$:

• The resulting terms after polynomial multiplication are $\$x^2\$$, $\$4x\$$, $\$3x\$$, and $\$12\$$.
• Here, $\$4x\$$ and $\$3x\$$ are like terms, as both have the variable $\$x\$$ raised to the same power.
• Adding these together gives $\$7x\$$, simplifying the expression to $\$x^2\; +\; 7x\; +\; 12\$$.

Understanding how to identify and combine like terms is crucial for anyone working to master algebraic expressions.