Problem 19 - Elementary Algebra

Question

Are the following factored correctly? Check by multiplying. $$ x 2+6 x+9=(x+3) 2 $$

Step-by-Step Solution

Answer

Yes, they are factored correctly.

Step 1: Expand the Factored Form

To verify if the factorization is correct, let's begin by expanding the right side of the equation \((x+3)^2\). This means we will multiply \((x+3)\) by itself.

Step 2: Apply the FOIL Method

Using the FOIL method (First, Outer, Inner, Last) on \((x+3)(x+3)\), we get:- **First:** \(x \cdot x = x^2\) - **Outer:** \(x \cdot 3 = 3x\) - **Inner:** \(3 \cdot x = 3x\) - **Last:** \(3 \cdot 3 = 9\) Combine these results for an intermediate expression.

Step 3: Combine Like Terms

After applying the FOIL method, the expression is: \[x^2 + 3x + 3x + 9\]Combine the like terms (\(3x + 3x\)):\[x^2 + 6x + 9\]

Step 4: Compare with Original Expression

Compare the expanded form \(x^2 + 6x + 9\) with the original expression \(x^2 + 6x + 9\). They are identical.

Key Concepts

FOIL Method

The FOIL method is a handy tool for multiplying two binomials. A binomial is a polynomial with two terms such as \((x + 3)\). When you multiply two binomials, you can use the FOIL method, which stands for First, Outer, Inner, Last, to keep track of all the terms to multiply.

• First: Multiply the first terms in each binomial. Here, multiply the \(x\) in \((x+3)\) and the \(x\) in the second \((x+3)\), leading to \(x^2\).
• Outer: Multiply the outer terms in the product. This involves the \(x\) from the first binomial and the \(3\) from the second, which gives \(3x\).
• Inner: Multiply the inner terms. Here it means multiplying \(3\) from the first binomial and \(x\) from the second, again getting \(3x\).
• Last: Multiply the last terms of each binomial. Multiply \(3\) and \(3\) to get \(9\).

The FOIL method helps ensure you don't miss any terms when expanding a product of binomials. It's especially useful when you're double-checking your work against an original equation.

Expand Binomials

Expanding binomials means expressing the product of two binomials as a single polynomial. This step involves revealing all terms that result from multiplying each term in one binomial with each term in another.

To expand \((x + 3)(x + 3)\) using the concepts from the FOIL method, multiply each term:

• First, multiply \(x\) and \(x\), resulting in \(x^2\).
• Next, from the Outer and Inner products, you'll have \(3x + 3x\), which represents combining the terms created by the different distributions.
• Finally, multiply the last terms, \(3\) and \(3\), yielding \(9\).

The expanded form from these steps is \(x^2 + 3x + 3x + 9\). This expression might feel a bit cluttered, but that’s where the concept of combining like terms comes in handy for simplifying.

Like Terms in Algebra

Like terms in algebra are terms that have the same variables raised to the same powers, even if their coefficients are different. When working with expressions, combining like terms simplifies the problem and makes the expression clearer.

In the expanded result from our work with \((x+3)(x+3)\), the expression is originally \(x^2 + 3x + 3x + 9\). Notice that \(3x\) and \(3x\) are like terms because they contain the same variable to the same power.

Combining like terms means adding together their coefficients:

• For \(3x + 3x\), add the coefficients 3 and 3 to get \(6x\).

This simplification transforms our expression to \(x^2 + 6x + 9\), aligning exactly with the original expression previously given, which signifies a correct factorization if that went unobserved at first.