Problem 21 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(x-6)^{2}$$

Step-by-Step Solution

Answer

The expanded form of \((x - 6)^2\) is \(x^2 - 12x + 36\).

Step 1 - Identify the Binomial to Expand

The task is to find the square of the binomial \((x - 6)\). This can be expanded using the formula for the square of a binomial, which is \((a - b)^2 = a^2 - 2ab + b^2\), where \(a\) is \(x\) and \(b\) is \(6\).

Step 2 - Apply the Binomial Square Formula

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), substitute \(a = x\) and \(b = 6\) into the equation: \[(x - 6)^2 = x^2 - 2(x)(6) + 6^2\]

Step 3 - Calculate Each Term

Compute each term from the expansion:- \(x^2\) remains as is.- \(-2(x)(6)\) simplifies to \(-12x\).- \(6^2\) equals \(36\).Thus, the expansion yields: \(x^2 - 12x + 36\).

Step 4 - Write the Final Expression

Combine all computed terms to write the final expression:\[x^2 - 12x + 36\]

Key Concepts

Binomial Theorem

The binomial theorem is a powerful algebraic tool that helps in expanding expressions like \((a + b)^n\). It's very useful when dealing with polynomials, where you multiply together terms and raise sums to powers. The general formula for the binomial theorem is:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, \( \binom{n}{k} \) represents the binomial coefficients, which tell us the number of ways to choose \( k \) elements from \( n \) without regard to order. These coefficients can be found in Pascal's Triangle or calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

It's crucial to understand this theorem, as it forms the basis for polynomial expansion and helps us easily find any term in the expanded form of a binomial.

Polynomial Expansion

Polynomial expansion is the process of multiplying and simplifying polynomials into an extended form. When dealing with binomial expressions like \((x-6)^2\), expanding means to rewrite the expression as a single polynomial.
Using formulas such as the square of a binomial can simplify this process. Expanding polynomials involves various techniques, including:

• Using distributive properties, where you distribute terms over each other.
• Applying specific binomial formulas, such as \((a + b)^2 = a^2 + 2ab + b^2\) or \((a - b)^2 = a^2 - 2ab + b^2\).
• Recognizing patterns and using shortcuts, like special products.

Polynomial expansion not only reveals the full expression but also simplifies solving equations or finding specific values.

Square of a Binomial

The square of a binomial is a specific form of polynomial expansion. It simplifies multiplying two identical binomials. For example, given a binomial \((x - 6)\):

• The square is \((x - 6)^2\).
• Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), with \(a\) as \(x\) and \(b\) as \(6\), we expand this expression.

Breaking it down:

• \(x^2\) results from squaring the first term.
• -2ab computes to -12x, where -12 is \(-2\times x\times 6\).
• The term \(6^2\), which is 36, from the square of the second term.

Therefore, the square of the binomial \((x-6)^2\) is \(x^2 - 12x + 36\). Working with binomials this way helps in quickly expanding and understanding the effects of operations on different algebraic expressions.