Problem 41 - Algebra for College Students

Question

Expand each binomial and simplify.\((-a+b)^{3}\)

Step-by-Step Solution

Answer

\((-a+b)^{3} = -a^3 + 3a^2b - 3ab^2 + b^3\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Use this theorem to expand \((-a+b)^3\). Here, \(x = -a\), \(y = b\), and \(n = 3\).

Step 2: Identify the Binomial Coefficients

For \(n = 3\), the binomial coefficients are \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\). These coefficients equal 1, 3, 3, and 1 respectively.

Step 3: Substitute Values into the Binomial Expansion

Substitute each of \(x = -a\) and \(y = b\) along with the binomial coefficients into the formula: \((-a+b)^3 = \binom{3}{0}(-a)^3b^0 + \binom{3}{1}(-a)^2b^1 + \binom{3}{2}(-a)^1b^2 + \binom{3}{3}(-a)^0b^3\).

Step 4: Simplify Each Term

Calculate each term separately:\[(-a)^3b^0 = -a^3,\]\[3(-a)^2b = 3a^2b,\]\[3(-a)b^2 = -3ab^2,\]\[1b^3 = b^3.\]

Step 5: Combine and Simplify the Expansion

Combine the simplified terms from Step 4 to get the expanded expression:\(-a^3 + 3a^2b - 3ab^2 + b^3\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in mathematics used to expand expressions in the form of \((x + y)^n\). This theorem allows us to break down a binomial raised to a power into a series of terms that can be more easily managed and solved. Based on the general formula:

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

For each term in the expansion:

• The coefficient is a binomial coefficient, which is calculated using combinations.
• The exponent for \(x\) decreases from \(n\) to 0, while the exponent for \(y\) increases from 0 to \(n\).

In our specific problem, we are dealing with the binomial \((-a + b)^3\). Here, \(x = -a\), \(y = b\), and \(n = 3\). We rely on the theorem to expand this binomial into a series of terms with coefficients derived from combination calculations.

Polynomial Simplification

Once we have expanded the binomial using the Binomial Theorem, the next step is Polynomial Simplification. Simplifying a polynomial means reducing it to its simplest form by combining like terms and arranging terms in a standard order. Here’s how you can think about it:Each term of the polynomial from the expansion is computed separately, considering:

• The coefficients.
• Powers of each variable.

This involves simplifying individual terms and then combining them.For instance, when we calculate each term of \((-a + b)^3\):

• We find \((-a)^3\) becomes \(-a^3\), \((3)(-a)^2b\) simplifies to \(3a^2b\).
• Similarly, \((3)(-a)b^2\) simplifies to \(-3ab^2\), and \(b^3\) remains as it is.

Finally, combining these terms, we arrive at the simplified polynomial form: \(-a^3 + 3a^2b - 3ab^2 + b^3\). This demonstrates the power of simplification in making complex expansions comprehensible.

Algebraic Expressions

Algebraic Expressions are expressions that contain variables, constants, and operators such as addition, subtraction, multiplication, and division. Understanding how to work with these expressions is fundamental to solving algebra problems.In the context of our exercise, each part of the solution revolves around manipulating algebraic expressions. Here are key points to consider:

• Identifying and using the variables \(a\) and \(b\) correctly.
• Performing operations on these variables based on the algebraic rules.

To exemplify, in \((-a + b)^3\), \(a\) and \(b\) are manipulated through powers and coefficients during expansion and simplification. Recognizing each term as an algebraic expression helps:

• Ensure we manage negative signs accurately.
• Keep track of like terms for combining them efficiently.

Mastery of algebraic expressions not only aids in binomial expansion but also in various other algebraic operations and is an essential skill for further algebraic learning.