Problem 69 - Algebra 2

Question

What is the expanded form of \((a-b)^{3} ?\) \(\begin{array}{ll}{\text { A. } a^{3}+a^{2} b+a b^{2}+b^{3}} & {\text { B. } a^{3}+3 a^{2} b+3 a b^{2}+b^{3}} \\ {\text { C. } a^{3}-a^{2} b+a b^{2}-b^{3}} & {\text { D. } a^{3}-3 a^{2} b+3 a b^{2}-b^{3}}\end{array}\)

Step-by-Step Solution

Answer

The expanded form of \((a-b)^{3}\) is \(a^{3} - 3a^{2}b + 3ab^{2} - b^{3}\). Therefore, the answer is D. a^{3}-3 a^{2} b+3 a b^{2}-b^{3}.

Step 1: Apply the binomial theorem

The binomial theorem states that \((x+y)^{n}=\sum _{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). Comparing this with the given \((a-b)^{3}\), we can consider \(a\) as \(x\), \(-b\) as \(y\), and \(n\) as \(3\)

Step 2: Substitute and simplify

Substitute the values into the binomial theorem formula: \((a-b)^{3}= \binom{3}{0} a^{3} (-b)^{0} + \binom{3}{1} a^{2} (-b)^{1}+ \binom{3}{2} a^{1} (-b)^{2}+ \binom{3}{3} a^{0} (-b)^{3}\). \nNext simplify the coefficients which are the binomial coefficients, and the powers of \(-b\): \((a-b)^{3}= 1*a^{3}*1 - 3*a^{2}*b + 3*a*b^{2} - 1*b^{3}\)

Step 3: Final simplified form

The final simplified form is \((a-b)^{3}=a^{3} - 3a^{2}b + 3ab^{2} - b^{3}\)

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that helps to expand expressions of the form \((x + y)^n\). It provides a formula to express these expansions as a sum of terms involving powers of \(x\) and \(y\). This theorem states:

• \((x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\)

In this formula, \(n\) is a non-negative integer, and \(\binom{n}{k}\) represents the binomial coefficients.
Each term of the expansion is obtained by multiplying the appropriate binomial coefficient by powers of \(x\) and \(y\) that add up to \(n\). This results in a polynomial of degree \(n\) with \((n+1)\) terms.
Applying this theorem to \((a-b)^3\) requires us to view \(-b\) as the second term and follow through with the expansion, keeping in mind the alternating signs that occur due to the negative sign.

Binomial Coefficients

Binomial coefficients are numerical factors that appear in the expansion given by the binomial theorem. They are represented by \(\binom{n}{k}\) and pronounced as "n choose k."
These coefficients are calculated using the formula:

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

where \(n!\) denotes the factorial of \(n\).
In simpler terms, the coefficients represent combinations of choosing \(k\) items out of \(n\) and are symmetric.
This means the coefficient for the \(k^{th}\) term is the same as for the \((n-k)^{th}\) term.
For the expression \((a-b)^3\), the coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\), giving the expanded terms with appropriate powers of \(a\) and \(-b\).

Polynomial Expansion

Polynomial expansion refers to the process of expressing a power of a binomial as a sum of terms. Each term in the expansion involves coefficients, which are the binomial coefficients, and variables raised to different powers.
When we expand \((a-b)^3\), we express it as a polynomial:

• First term: \(a^3\)
• Second term: \(-3a^2b\)
• Third term: \(3ab^2\)
• Fourth term: \(-b^3\)

The sign alternates due to the negative second term \(-b\) in each combination, which affects how powers of \(-b\) are expressed.
The purpose of using polynomial expansion in binomials is to enable calculations, simplifications, and manipulations of more complex algebraic expressions. This method can be applied to any polynomial power, making algebraic calculations more feasible.