Problem 63 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (x+2)(x-2) $$

Step-by-Step Solution

Answer

The result is \(x^2 - 4\).

Step 1: Recognize the Formula

Observe that the expression \((x+2)(x-2)\) is a product of binomials in the form \((a+b)(a-b)\). This specific format suggests the use of the difference of squares formula.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\). Here, \(a = x\) and \(b = 2\). Substitute these values into the formula.

Step 3: Perform the Substitution

Substitute \(x\) for \(a\) and \(2\) for \(b\) in the formula: \(x^2 - 2^2\).

Step 4: Simplify the Expression

Calculate \(2^2 = 4\) and simplify the expression to \(x^2 - 4\).

Step 5: Verify the Solution

Double-check that the expression \(x^2 - 4\) is indeed the expansion of \((x+2)(x-2)\). This can be verified by multiplying out the original expression: \((x+2)(x-2) = x^2 - 2x + 2x - 4\), which simplifies back to \(x^2 - 4\) since the middle terms cancel out.

Key Concepts

Binomials

Binomials are special kinds of algebraic expressions. They are called 'binomials' because 'bi' means two, and 'nomial' refers to terms. So, in simple terms, a binomial is a polynomial with exactly two terms. For example,

• \(x + 2\)
• \(x - 2\)

are both binomials. Binomials can be added, subtracted, and multiplied just like numbers. When multiplying binomials, a special case occurs that is useful in algebra: the difference of squares. Recognizing this form can make simplification faster and easier. The form

• \((a+b)(a-b)\)

is particularly notable because it simplifies into a difference of two squares, \(a^2 - b^2\). This pattern often appears in quadratic equations and advanced algebraic operations, and being familiar with it can help solve problems quickly and accurately.

Multiplication of Polynomials

Multiplying polynomials involves using the distributive property repeatedly. When we multiply two binomials, like \((x+2)(x-2)\), we consider each term in the first binomial multipled by each term in the second. For our example, this involves four multiplications:

• \(x \times x\)
• \(x \times -2\)
• \(2 \times x\)
• \(2 \times -2\)

This multiplication approach is the basis for expanding binomials. However, when the binomials are in the format of \((a+b)(a-b)\), we recognize it follows the difference of squares pattern. This means we don’t need to manually go through each multiplication step, saving time and potential for error. Applying the formula \[\(a^2 - b^2\)\] directly gives us the result because the middle terms \(-2x\) and \(+2x\) cancel each other out.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. This often means combining like terms and applying known algebraic identities, such as the difference of squares. For example, after applying the difference of squares formula to \((x+2)(x-2)\), we end with \(x^2 - 4\). The middle terms \(-2x\) and \(+2x\) cancel each other, simplifying our job. Simplifying expressions not only makes equations easier to work with but is also crucial when solving and understanding algebraic equations. In practice, always try to identify patterns or formulas that can simplify your work. Checking the final simplified expression by multiplying it back to reconstruct the original expression ensures accuracy. In this case, multiplying \(x^2 - 4\) gives us the same terms as the original multiplication of \((x+2)(x-2)\), confirming the simplification process was done correctly.