Question

Use a vertical format or a horizontal format to add or subtract. $$\left(u^{3}-u\right)-\left(u^{2}+5\right)$$

Short answer

The simplified expression in either vertical format or horizontal format after subtraction is \(u^{3}-u^{2}-u-5\).

Step-by-step solution

Distribute the Negative Sign

Distribute the negative sign, which is implied by the subtraction operator, to all the terms of the second polynomial expression. This changes the subtraction problem into an addition problem, which is easier to solve. The expression becomes: \(u^{3}-u- (u^{2}+5)\).

Simplify the Polynomial

Combine like terms by adding or subtracting the coefficients (numerical parts) of the variables that have the same exponent. The expression becomes: \(u^{3}-u^{2}-u-5\).

Verify and Write the Final Equation

Ensure that the final equation presented is the simplest form of the equation that can't be further reduced. The final equation here is: \(u^{3}-u^{2}-u-5\).

Key concepts

Combining Like Terms

When working with polynomials, one of the essential steps to simplifying the expression is to \textbf{combine like terms}. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(3x^2 + 4x - x^2 + 2x\), the terms \(3x^2\) and \( -x^2\) are like terms, and so are \(4x\) and \(2x\).

To combine them, you simply add or subtract the coefficients (the numbers before the variable). It's like saying you have 3 apples (\(3x^2\)) and you take away one apple (\( -x^2\)), leaving you with 2 apples (\(2x^2\)). Similarly, if you have 4 oranges (\(4x\)) and add 2 more oranges (\(2x\)), you end up with 6 oranges (\(6x\)). So, the expression simplifies to \(2x^2 + 6x\).

Applying this to our given example, we look for terms with the same exponent of \(u\). There are none to combine, which means each term is unique in this case. Therefore, the simplified expression remains unchanged as \(u^{3}-u^{2}-u-5\).

Distributing Negative Sign

The process of \textbf{distributing a negative sign} across an expression is a critical step in polynomial subtraction. When faced with a problem like \(\left(u^{3}-u\right)-\left(u^{2}+5\right)\), that negative sign before the second group of parentheses is essentially a \( -1 \) multiplying everything inside those parentheses.

So, what does it mean to distribute the negative sign? Imagine you owe someone 5 dollars; represented by \( -5 \), and you owe one more dollar, represented by \( -1 \), the total amount you owe is the sum: \( -5 - 1 = -6 \). Distributing the negative sign works the same way: \(\left(u^{3}-u\right) - (u^{2}+5)\) becomes \(u^{3} - u - u^{2} - 5\) when the negative sign is distributed. Each term inside the parentheses is now subtracted, changing the entire dynamic of the equation.

This allows us to convert a subtraction problem into the much more manageable process of addition, leading us smoothly to the next step: combining like terms and reaching the final simplified form of the polynomial.

Simplifying Expressions

Simplifying expressions is the last stage in the journey of polynomial operations and is like cleaning up after cooking a meal — you want to end up with a clean and clear kitchen, or in this case, a clear mathematical equation.

In simplifying an expression, every effort is made to reduce the equation to its simplest form. This involves combining like terms, as we previously discussed, and ensuring that no further operations can be carried out on the polynomial. \textbf{Simplifying expressions} may also include factoring, if possible, or even canceling out terms if they appear on both sides of an equation.

In our exercise, \(u^{3}-u^{2}-u-5\) is already in its simplest form. None of the terms can be combined with another, and there are no common factors to extract. It's crucial to verify this, as simplifying ensures clarity and can sometimes reveal hidden patterns or solutions that were not initially apparent. The simplicity of the final expression makes it easier to work with in subsequent calculations or algebraic manipulations.