Expanding Algebraic Expressions: The Difference of Squares

The question asks for the expansion of the algebraic expression \( (x + 11)(x - 11) \). This expression is in a specific form that can be expanded using a common algebraic identity. Recognizing this pattern is key to quickly finding the correct expansion.

The form of the expression \( (x + 11)(x - 11) \) matches the algebraic identity known as the difference of squares.

The difference of squares identity states:

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

In the given expression \( (x + 11)(x - 11) \), we can see that:

• \( a \) corresponds to \( x \)
• \( b \) corresponds to \( 11 \)

Now, we can apply the difference of squares identity by substituting \( a = x \) and \( b = 11 \) into the formula \( a^2 - b^2 \):

Expansion \( = x^2 - 11^2 \)

Next, we need to calculate the value of \( 11^2 \).

\( 11^2 = 11 \times 11 = 121 \)

Substituting this value back into the expansion:

Expansion \( = x^2 - 121 \)

Therefore, the expansion of \( (x + 11)(x - 11) \) is \( x^2 - 121 \).

Alternative Method: Using FOIL

If you don't immediately recognize the difference of squares identity, you can also expand the expression using the FOIL method (First, Outer, Inner, Last). This method involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms.

For \( (x + 11)(x - 11) \):

• First: Multiply the first terms: \( x \times x = x^2 \)
• Outer: Multiply the outer terms: \( x \times (-11) = -11x \)
• Inner: Multiply the inner terms: \( 11 \times x = +11x \)
• Last: Multiply the last terms: \( 11 \times (-11) = -121 \)

Now, combine these results:

\( x^2 - 11x + 11x - 121 \)

Combine the like terms (\( -11x \) and \( +11x \)):

\( -11x + 11x = 0x = 0 \)

So the expression simplifies to:

\( x^2 + 0 - 121 = x^2 - 121 \)

Both the difference of squares identity and the FOIL method give the same result, \( x^2 - 121 \). The difference of squares identity is often faster when applicable.

Analyzing the Options

Let's look at the given options and compare them to our result \( x^2 - 121 \).

Based on our expansion, the correct option is \( x^2 - 121 \).

Revision Table: Key Concepts in Expansion

Additional Information: Important Algebraic Identities

Besides the difference of squares, there are other fundamental algebraic identities that are very useful for expanding and factoring expressions. Knowing these identities can save time and help simplify calculations in algebra.

• Square of a Sum: \( (a + b)^2 = a^2 + 2ab + b^2 \)
• Square of a Difference: \( (a - b)^2 = a^2 - 2ab + b^2 \)
• Sum of Cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
• Difference of Cubes: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

Mastering these identities is essential for solving various problems in algebra, including factoring polynomials, solving equations, and simplifying complex expressions. The expansion of \( (x + 11)(x - 11) \) is a direct application of one of the most common identities, the difference of squares.