Problem 73 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to expand \(\left(x^{3}-y^{4}\right)^{5},\) I find it helpful to rewrite the expression inside the parentheses as \(x^{3}+\left(-y^{4}\right)\)

Step-by-Step Solution

Answer

The statement makes sense, as the subtraction of a term can indeed be rewritten as the addition of its negation, and this does not affect the expansion of the expression.

Step 1: Minus to Plus

The given expression \(\left(x^{3}-y^{4}\right)^{5}\) has been rewritten into the form of \(x^{3}+\left(-y^{4}\right)\). This is based on the fact that subtracting a term is the same as adding its negation.

Step 2: Expansion Property Validation

In both the given and rewritten forms, the expressions within the parentheses are being raised to the power of 5. According to the properties of arithmetic, changing the subtraction between two terms to an addition of the negative of a term does not affect its expansion.

Step 3: Conclusion

Therefore, the statement makes sense because rewriting the expression inside the parentheses as \(x^{3}+\left(-y^{4}\right)\) does not affect the expansion of the whole expression \(\left(x^{3}-y^{4}\right)^{5}\).

Key Concepts

Properties of Arithmetic

The properties of arithmetic are the foundational rules for manipulating numbers and expressions in mathematics. Among these properties are the commutative, associative, distributive, identity, and inverse properties.

Understanding these properties allows you to rearrange and simplify expressions without changing their values. For instance, the commutative property implies that numbers can be added or multiplied in any order, whereas the associative property allows you to group numbers differently while adding or multiplying.

The distributive property is especially crucial when expanding polynomials. It states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results, exemplified by the formula: \( a(b+c) = ab + ac \). The identity properties refer to the zero and one as the additive and multiplicative identities, respectively, ensuring expressions remain unchanged when zero is added or any number is multiplied by one. Conversely, the inverse properties pertain to the idea of negation and reciprocation, allowing for the simplification of expressions through the subtraction of a number or division by it.

By applying these properties, especially when dealing with operations such as rewriting subtraction as the addition of a negative, as seen in the exercise, you ensure that equivalent expressions are generated, adhering to the rules that govern arithmetic operations.

Polynomial Expansion

Polynomial expansion is a process that involves applying arithmetic properties to multiply and simplify expressions with multiple terms – for example, distributing and combining like terms.

Expanding a polynomial like \( (x + y)^n \) requires using the binomial theorem or the distributive property repeatedly. This can also be achieved through a method called FOIL (First, Outer, Inner, Last) when dealing with binomials. In our example from the exercise, the expression \( (x^{3} - y^{4})^5 \) is raised to the fifth power, which involves significant expansion.

Understanding the Expansion Process

Note that when you raised the expression to a power, you multiply the expression by itself that many times. During expansion, each term in the polynomial gets combined with every other term, each time according to the exponent's value. This operation adheres to all properties of arithmetic.

For simpler expressions, polynomial expansion might be straightforward, but as the number of terms and the exponent value increase, the process can become complex. Hence, practicing the polynomial expansion methods and understanding their connection to arithmetic properties is crucial for mastering this skill.

Negative Exponents

Negative exponents play a unique role in the realm of arithmetic, particularly when they appear in polynomial expressions. The key principle behind negative exponents is that they represent the reciprocal of the base raised to the opposite positive exponent. Specifically, \( a^{-n} = \frac{1}{a^n} \), where \( a\) is not equal to zero.

This concept is frequently observed when dealing with polynomial expressions where subtraction is present, often leading students to misunderstand the correct approach for expansion.

Application in Polynomial Expansion

When you encounter a term with a negative exponent during expansion, remember to apply the negative exponent rule, which converts the negative exponents into positive ones by placing them in the denominator of a fraction. In the context of the exercise, rewriting \( -y^{4} \) as \( (-1)y^{4} \) makes it clearer that the negative sign is associated with a coefficient of \( y^{4} \) and is to the first power (since \( -1 = (-1)^1 \) with a negative exponent of 1).

Applying your understanding of negative exponents correctly ensures that you can manipulate and simplify polynomial expressions accurately, which is essential in more advanced algebra and calculus work. Regular practice with these concepts helps to solidify comprehension and ability to apply them in various mathematical scenarios.