Problem 45 - College Algebra

Question

Find each product. $$(x-3)^{2}$$

Step-by-Step Solution

Answer

The product upon expanding \((x-3)^2\) is \(x^2 - 6x + 9\).

Step 1: Write Out the Expression

Write out the expression \((x-3)^2\). We should remember that \((x-3)^2\) is the same as \((x-3)\) times \((x-3)\). Therefore, let's rewrite the expression as: \((x-3)(x-3)\).

Step 2: Apply FOIL Method

Apply the FOIL (First, Outer, Inner, Last) method. This method helps to simplify the multiplication of two binomials. Here, \n1. First multiply \(x\) (the first term of the first binomial) by \(x\) (the first term of the second binomial) to get \(x^2\).\n2. Then, multiply \(x\) (the first term of the first binomial) by \(-3\) (second term of the second binomial) to get \(-3x\) (Outer).\n3. Multiply \(-3\) (the second term of the first binomial) by \(x\) (first term of the second binomial) to get \(-3x\) (Inner).\n4. Finally, multiply \(-3\) (second term of the first binomial) by \(-3\) (second term of both binomials) to get \(9\) (Last).\n\nAdd all these results together - \(x^2, -3x, -3x, 9\).

Step 3: Simplify the Result

Combine like terms. \(-3x - 3x\) combine to \(-6x\). So, rewrite the equation as \(x^2 - 6x + 9\)

Key Concepts

The FOIL Method in Algebra

When multiplying binomials, the FOIL method is a straightforward approach to ensure you account for each term. The name FOIL stands for First, Outer, Inner, Last, which refers to the position of each term in the binomials being multiplied. Let's consider the given exercise where we need to find the product of \( (x-3)^2 \). This expression can be understood as \( (x-3)(x-3) \). We multiply the First terms, Outer terms, Inner terms, and Last terms respectively.

For our example, the First terms are both \(x\)'s, thus multiplying them gives us \(x^2\). The Outer terms are \(x\) and \( -3 \), leading to \( -3x \). The Inner multiplication also involves \( -3 \), but mirrors the Outer step, yielding an identical \( -3x\). Lastly, the Last terms which are both \( -3\)'s multiplied together give us a positive \(9\), because a negative times a negative is a positive. Our FOIL result is the sequence of terms: \(x^2, -3x, -3x, 9\).
Applying the FOIL method properly ensures that all possible products of terms are accounted for and is an essential skill in binomial multiplication.

Simplifying Algebraic Expressions

After applying the FOIL method in our given exercise to get the terms \(x^2, -3x, -3x, 9\), we move into simplifying the algebraic expression. Simplifying involves combining like terms and organizing the polynomial in a standard form, usually descending powers of \(x\). Like terms have the same variable raised to the same power. In this case, the \( -3x\) terms from the Outer and Inner multiplication steps are like terms. We combine them by adding the coefficients of each term together, which for our expression is \( -3 - 3 \) giving us \( -6x \).

So, our simplified expression becomes \(x^2 - 6x + 9\). This is not just about cleanliness or aesthetics; it's the conventionally accepted form for writing polynomials and is essential for further algebraic manipulation, such as solving equations or factoring.

Combining Like Terms

The concept of combining like terms is key in algebra to simplify expressions to their most basic form. Like terms, as the name implies, share the same variable(s) and the same exponents for those variables. By combining them, we're essentially doing basic arithmetic with their coefficients. To do this effectively, one must identify terms with the same variable(s) and exponent(s) and then add or subtract the coefficients.

In the final step of the exercise above, we combine the like terms \( -3x \) and \( -3x \), which come from the multiplication of the Outer and Inner terms of the binomials. Since the terms are identical in their variable and exponent, we simply add the coefficients, giving us \( -6 \) for the combined term of \( -6x\). This combined form, \( x^2 - 6x + 9\), is much simpler and easier to use in subsequent calculations or equations. It's vital for students to practice recognizing and combining like terms accurately, as it's a foundation for more advanced algebraic concepts.