91Ó°ÊÓ

When we talk about the binomial theorem, we are diving into the world of polynomials. A binomial is simply an algebraic expression containing two terms. Think about expressions like \( (x + 3) \) or \( (x - 13) \). The binomial theorem gives us a way to expand these expressions when they are raised to a power.
In our exercise, we have \( (x - 13)^2 \). This is a binomial raised to the power of 2. To expand it, we use a specific formula from the binomial theorem known as the square of a binomial formula:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)

Here, instead of adding \(a\) and \(b\), in our exercise, we are subtracting: \( (x - 13)^2 \). That means:

• \( a = x \)
• \( b = -13 \)

Using this formula, the expansion becomes \( x^2 + 2(x)(-13) + (-13)^2 \). It's like following a simple recipe for expansion!

Like terms in algebra are terms that have exactly the same variables raised to the same power. They are very crucial in simplifying expressions. For example, in the expression \( 2x - 3x + 5 \), \( 2x \) and \( -3x \) are like terms because they both contain the variable \( x \) raised to the same power.
In our expanded expression, \( x^2 - 26x + 169 \), we must look out for any like terms to further simplify it. Although we have terms with \( x^2 \), \( x \), and constants here, none of them are like one another:

• \( x^2 \) is a term with the variable raised to the second power.
• \( -26x \) is a term with the variable \( x \).
• \( 169 \) is a constant term with no variable.

As these are not like terms, they remain as they are in the expanded and simplified form.

Simplifying expressions makes understanding and working with them easier. When you simplify, you're basically combining like terms and clearing any unnecessary calculations.
In this exercise, once we expanded \( (x - 13)^2 \) to \( x^2 - 26x + 169 \), simplifying it involved:

• Carrying out the multiplication correctly: calculate \(-26x\) from \(2(x)(-13)\).
• Computing the square of the constant term: \((-13)^2\) gives us \(169\).

After performing these calculations, the expression \( x^2 - 26x + 169 \) cannot be simplified further because all terms are unlike. Thus, our focus was making sure each part of the expansion was calculated correctly, ensuring no further simplification was needed.