Problem 80 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There are no values of \(a\) and \(b\) such that $$ (a+b)^{4}=a^{4}+b^{4} $$

Step-by-Step Solution

Answer

The statement is false. The correct formulation following the binomial theorem would be \((a+b)^{4}=a^{4} + 4a^3b + 6a^2b^2 + 4ab^3 + b^{4}\)

Step 1: Recognize The Binomial Theorem

The Binomial theorem describes the expansion of powers of a binomial. It states that: \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k}\), where \({n \choose k}\) denotes the number of ways to choose \(k\) from \(n\), also known as the binomial coefficient.

Step 2: Apply the Binomial Theorem to the Equation

Expand the left side of our original equation \((a+b)^4\) using the Binomial theorem. You obtain \((a+b)^4 = {4 \choose 0} a^4 + {4 \choose 1} a^3b + {4 \choose 2} a^2b^2 + {4 \choose 3} ab^3 + {4 \choose 4} b^4\). Simplifying this results in \(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\).

Step 3: Compare the Expanded Form to the Original Statement

Comparing this to the original statement \(a^4 + b^4\), it's clear that the two are not the same. The original statement is missing three of the terms from the expanded form. Thus, the initial statement is false.

Step 4: Formulate the corrected statement

A correct statement would be \((a+b)^{4}=a^{4} + 4a^3b + 6a^2b^2 + 4ab^3 + b^{4}\), as derived from the Binomial theorem.

Key Concepts

Polynomial Expansion

When working with binomials raised to a power, we're often dealing with polynomial expansion. This is the process of expanding an expression like \((a+b)^n\) into a sum of terms involving powers of \(a\) and \(b\). The Binomial Theorem provides a systematic way to perform this expansion of a polynomial.The theorem states that for any non-negative integer \(n\), the expansion is \[(a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k}\].The binomial coefficients \({n \choose k}\) indicate how many ways you can choose \(k\) elements from \(n\) elements without regard to order.Polynomial expansion is a powerful tool as it helps in simplifying expressions and solving equations, especially in algebra.

• It provides a shortcut over multiplying the binomial repeatedly.
• It helps in understanding how distributions form across different terms.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations such as addition or subtraction. They are fundamental in mathematics because they represent relationships that can be manipulated using algebraic laws.An important aspect in algebra is the simplification of expressions. When you apply the Binomial Theorem to expand expressions such as \((a+b)^4\), you are expressing the equation in terms of polynomial series like a^4, 4a^3b, 6a^2b^2, 4ab^3, and b^4.In the context of the given exercise, the goal was to identify whether an expanded expression could be equal to its simplified terms.

• Simplification makes equations easier to solve or compare.
• Proper expansion ensures all terms are accounted for, essential in correct formulation.

False Statements in Mathematics

Encountering false statements in mathematics can often be a learning point. It involves investigating why the statement doesn't hold true and correcting any misconceptions.In our example, the false statement was thinking that \((a+b)^4\) could be simply expressed as \(a^4 + b^4\). By expanding using the Binomial Theorem, we found additional terms that were missing initially.Correcting such false statements:

• Strengthens understanding of mathematical rules such as the Binomial Theorem.
• Enhances problem-solving skills by encouraging critical analysis and deduction.

Recognizing and addressing these falsehoods not only helps in solidifying algebraic concepts but also teaches vigilance in ensuring mathematical accuracy.