Problem 52 - Elementary Algebra

Question

\(\left(w^{3}-w-5\right)-\left(w^{3}-w^{2}-5\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(ewline w^{2} - wewline \).

Step 1 - Write down the original expression

Start by writing the given expression: ewline ewline \(ewline (w^{3}-w-5)-(w^{3}-w^{2}-5)ewline \)

Step 2 - Distribute the negative sign

Distribute the negative sign to each term inside the parentheses: ewline ewline \(ewline w^{3} - w - 5 - w^{3} + w^{2} + 5ewline \)

Step 3 - Combine like terms

Now, combine the like terms by adding or subtracting them: ewline ewline Combine \(ewline w^{3} - w^{3}ewline \), \(ewline -w + w^{2}ewline \), and \(ewline -5 + 5ewline \). This simplifies to: ewline ewline \(ewline 0 + w^{2} - w + 0ewline \)

Step 4 - Write the final simplified expression

Write down the resulting simplified expression: ewline ewline \(ewline w^{2} - wewline \)

Key Concepts

Polynomial Expressions

A polynomial is an expression made up of variables, coefficients, and exponents combined with addition, subtraction, and multiplication. Polynomials can have one term, multiple terms, or even no variables at all. For example, in the expression: \( w^3 - w - 5 \) * Each term is separated by a plus or minus sign. * 'w^3' is a term with a variable 'w' raised to the power of 3. * '-w' is another term with the variable 'w' raised to the power of 1. * '-5' is a term without any variables, also known as a constant.Polynomials can be simplified and manipulated in various ways, including addition, subtraction, multiplication, and division.

Like Terms

Like terms are terms in a polynomial that have the same variable raised to the same power. When simplifying polynomials, it is important to combine like terms. Consider \( w^3 - w - 5 - ( w^3 - w^2 - 5 ) \).First, distribute the negative sign so it becomes:\( w^3 - w - 5 - w^3 + w^2 + 5 \).Then, combine like terms: * Combine \( w^3 \text{ and } -w^3 \) to get 0.* Combine \( -w \text{ and } +w^2 \), which results in \( w^2 - w \).By identifying and combining like terms, we effectively simplify the polynomial: \( w^2 - w \).

Distributive Property

The distributive property is a useful mathematical rule applied to expressions inside parentheses. It states that a term outside the parentheses should be multiplied by each term inside the parentheses:\( a(b + c) = ab + ac \).In polynomial subtraction, this property helps distribute the negative sign. Take the expression \(( w^3 - w - 5 ) - ( w^3 - w^2 - 5 ) \).Distribute the subtraction:\( w^3 ) - w - 5 - w^3 + w^2 + 5 \).Notice how each term in the second polynomial \( - ( w^3 - w^2 - 5 ) \)changes its sign when the subtraction is distributed. This helps when combining like terms and ultimately simplifies the polynomial:\( w^2 - w \). Understanding and applying the distributive property makes polynomial operations more straightforward.