Problem 21 - Elementary Algebra

Question

For the following problems, find the products. $$ (x-12)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of the expression \((x-12)^2\) is \(x^2 - 24x + 144\).

Step 1: Understanding the Expression

The given expression is \((x-12)^2\). This means that we need to multiply the binomial \((x-12)\) with itself: \((x-12)(x-12)\).

Step 2: Apply the Distributive Property (FOIL)

We will use the distributive property to multiply the terms of each binomial. This technique is also known as FOIL (First, Outer, Inner, Last). We follow the steps below: F - Multiply the first terms: \((x)(x) = x^2\) O - Multiply the outer terms: \((x)(-12) = -12x\) I - Multiply the inner terms: \((-12)(x) = -12x\) L - Multiply the last terms: \((-12)(-12) = 144\)

Step 3: Combine like terms and write the final expression

Now we need to combine the like terms and write the final expression: \(x^2 - 12x - 12x + 144\) Combine the like terms: \(x^2 - 24x + 144\) The product of the given expression \((x-12)^2\) is: $$ (x-12)^2 = x^2 - 24x + 144 $$

Key Concepts

Binomial Expansion

Binomial expansion is a mathematical process used to expand expressions that are raised to a power, typically involving two terms. The expression \( (x-12)^2 \) is a perfect example of binomial expansion, as it contains two terms (\( x \) and \( -12 \)) inside the parentheses. To expand such expressions, we repeat the multiplication of the binomial itself. This means, we consider \( (x-12) \) multiplied by \( (x-12) \).

Binomial expansion allows you to systematically approach the multiplication of these terms without confusion. When dealing with powers, each term from the first binomial multiplies every term of the second. This approach ensures that all combinations of terms are accounted for in the expansion process. With practice, using binomial expansion can be a straightforward way to handle higher powers as well.

Distributive Property

The distributive property is a key algebraic rule that further assists in breaking down expressions for easier computation. In multiplication of polynomials, this property gives a clear procedure to ensure every term in the first binomial is multiplied by each corresponding term in the second binomial.

• For the problem \( (x-12)(x-12) \), apply it by treating each term in the binomials separately.
• This means that the term \( x \) from the first binomial is multiplied with both terms \( x \) and \( -12 \) from the second.
• Similarly, \( -12 \) needs to multiply the terms \( x \) and \( -12 \) from the counterpart binomial.

The use of the distributive property guarantees that every interaction between terms is captured in the multiplication process. It's crucial to handle these steps carefully because forgetting even a single term can lead to incorrect results. With each multiplication, you work towards the complete expanded polynomial form.

FOIL Method

The FOIL method is a specific application of the distributive property used exclusively in multiplying two binomials. FOIL is an acronym that stands for First, Outer, Inner, Last, describing precisely the order of term multiplication.

This method works as follows when multiplying \( (x-12)(x-12) \):

• First: Multiply the first terms in each binomial. Here, \( x imes x = x^2 \).
• Outer: Multiply the outer terms of each binomial. This step involves \( x imes -12 = -12x \).
• Inner: Multiply the inner terms. Thus, \( -12 imes x = -12x \).
• Last: Multiply the last terms in each binomial, getting \( -12 imes -12 = 144 \).

After performing these steps, the next phase is to combine like terms, simplifying the expression into its final polynomial form. Here, like terms are combined to yield \( x^2 - 24x + 144 \). This method is particularly popular because it straightforwardly organizes multiplication of binomials, aiding students in following a logical sequence.