Problem 99 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(k-2)^{2}$$

Step-by-Step Solution

Answer

The special product of \((k-2)^2\) is \(k^2 - 4k + 4\).

Step 1: Identify the values of \(a\) and \(b\) #

In the expression \((k-2)^2\), we have \(a=k\) and \(b=2\).

Step 2: Apply the formula \((a-b)^2 = a^2 - 2ab + b^2\) #

Substituting the values of \(a\) and \(b\) in the formula, we get: \[(k-2)^2 = k^2 - 2\cdot k \cdot 2 + 2^2\]

Step 3: Simplify the expression #

Now, we will simplify the expression: \[(k-2)^2 = k^2 - 4k + 4\] Hence, the special product of \((k-2)^2\) is \(k^2 - 4k + 4\).

Key Concepts

Expanding Binomials

Expanding binomials is a simple but important algebraic process that involves using a formula to multiply two terms within parentheses. Binomials are expressions with two terms separated by a plus or minus sign, and they form the basis of more complex polynomial expressions.
For example, when you see something like

• \((k-2)^2\)

it means you need to multiply

• \((k-2)\) by \((k-2)\)

Expanding this binomial requires recognizing that it is a special product called the square of a binomial. The formula to expand

• \((a-b)^2\)
• \((a+b)^2\)

is

• \((a-b)^2 = a^2 - 2ab + b^2\)
• \((a+b)^2 = a^2 + 2ab + b^2\)

Using these formulas, students can quickly expand the binomial and arrive at a much simpler expression.

Polynomial Expressions

Polynomial expressions are foundational in algebra, consisting of terms made up of variables and coefficients. They can be as simple as a single term or a combination of several terms connected by addition or subtraction.
In the case of expanding binomials, you end

• with a polynomial that is the result of multiplication.
• For instance,
• \((k-2)^2\) becomes \(k^2 - 4k + 4\), which is a polynomial with three terms.

When working with polynomials, it is important to remember to

• collect like terms,

meaning any terms that have the same power of the variable.
This simplifies and organizes the expression, making it easier to understand and use in further calculations.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. They offer shortcuts and techniques to simplify algebraic problems, particularly in expanding and factoring expressions.
One of the most common algebraic identities is the binomial square identity, given as

• \((a-b)^2 = a^2 - 2ab + b^2\).

Recognizing and applying identities like this can help students to expand binomials with confidence.
This identity ensures that when you see

• \((k-2)^2\),

you can expand it easily into

• \(k^2 - 4k + 4\).

Remember that these identities are useful tools that simplify complex problems into manageable steps. By practicing them, students can enhance their algebraic skills and solve equations more efficiently.