Problem 26 - A Graphical Approach to College Algebra

Question

Write the binomial expansion for each expression. $$(a-b)^{7}$$

Step-by-Step Solution

Answer

The binomial expansion is \(a^7 - 7a^6b + 21a^5b^2 - 35a^4b^3 + 35a^3b^4 - 21a^2b^5 + 7ab^6 - b^7\).

Step 1: Identify the Binomial Coefficients

To expand \((a-b)^7\) , we use the binomial theorem, which states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\) In your case, replace \(b\) with \(-b\). For each term, the coefficient is given by the binomial coefficient \(\binom{7}{k}\), where \(k\) ranges from 0 to 7.

Step 2: Compute the Binomial Coefficients

Calculate the binomial coefficients for \(k = 0\) to \(7\):- \(\binom{7}{0} = 1\)- \(\binom{7}{1} = 7\)- \(\binom{7}{2} = 21\)- \(\binom{7}{3} = 35\)- \(\binom{7}{4} = 35\)- \(\binom{7}{5} = 21\)- \(\binom{7}{6} = 7\)- \(\binom{7}{7} = 1\)

Step 3: Construct the Terms of the Expansion

Substitute \((a-b)\) in the binomial theorem:Each term given by the binomial theorem:a) \((a-b)^7 = \sum_{k=0}^{7} \binom{7}{k} a^{7-k} (-b)^k\)b) Plug in the computed coefficients and simplify:1. \(\binom{7}{0}a^7(-b)^0 = a^7\)2. \(\binom{7}{1}a^6(-b)^1 = -7a^6b\)3. \(\binom{7}{2}a^5(-b)^2 = 21a^5b^2\)4. \(\binom{7}{3}a^4(-b)^3 = -35a^4b^3\)5. \(\binom{7}{4}a^3(-b)^4 = 35a^3b^4\)6. \(\binom{7}{5}a^2(-b)^5 = -21a^2b^5\)7. \(\binom{7}{6}a^1(-b)^6 = 7ab^6\)8. \(\binom{7}{7}a^0(-b)^7 = -b^7\)

Step 4: Write the Full Expansion

Combine all terms from Step 3 to write the expanded form:- \(a^7 - 7a^6b + 21a^5b^2 - 35a^4b^3 + 35a^3b^4 - 21a^2b^5 + 7ab^6 - b^7\)

Key Concepts

Binomial Theorem

The binomial theorem is a vital concept in algebra that provides us with a way to expand expressions raised to a power, specifically binomials. A binomial is an algebraic expression that consists of two terms, like \((a + b)\). The binomial theorem gives us a formula to expand these expressions efficiently without multiplying them manually, which can be cumbersome for large exponents. The theorem is expressed as: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \] What this means is that instead of multiplying \((a+b)\) by itself \(n\) times, we can use the summation notation \(\sum\) and binomial coefficients \(\binom{n}{k}\) to find each term of the expansion directly. Each term involves the binomial coefficient, a power of \(a\), and a power of \(b\). The index \(k\) starts from 0 and increases to \(n\), which ensures that every single term from \(a^n\) to \(b^n\) is covered.

Binomial Coefficients

In the binomial theorem, binomial coefficients \(\binom{n}{k}\) play a crucial role. They determine the coefficients for each term in the expanded form of a binomial expression. These coefficients are derived from Pascal’s Triangle or calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here's how they work:

• This formula uses factorials, which are the product of an integer and all the integers below it. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• The value of \(n\) is the exponent in the binomial expression \((a+b)^n\).
• \(k\) is the term position number in the expansion, which runs from 0 to \(n\).

These coefficients have important properties, such as symmetry. For example, \(\binom{7}{2} = \binom{7}{5}\). This is because in a row of Pascal's Triangle, coefficients are mirror images around the middle.

Algebraic Expressions

An algebraic expression, such as \((a-b)^7\), involves numbers, variables, and arithmetic operations (addition, subtraction, multiplication, etc.). The challenge often lies in simplifying expressions to reveal their expanded form, especially when it involves powers. Let's consider our binomial expression: To expand \((a-b)^7\), we use the binomial theorem by substituting \(b\) with \(-b\). This lets us apply the formula to get each term of the expansion:

• We handle each term \((a-b)^k\) separately and compute its contribution using the pre-calculated binomial coefficients.
• This substitution alters the addition to subtraction in the formula, because of the negative sign.

Thus, algebraic expressions often require careful manipulation to maintain accuracy in results. Remember:- Simplify step-by-step.- Consider the influence of negative signs.- Verify each term through substitution and calculation.