Problem 48 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Simplified expression after subtraction is: \( u^{3} - u^{2} - u - 5 \)

Step 1: Rewrite the Expression

Start by rewriting the expression as follows and remove the parentheses: \( u^{3} - u - u^{2} - 5 \)

Step 2: Group the like terms together

Rearrange the terms so that same degree terms are together. The equation will become: \( u^{3} - u^{2} - u - 5 \)

Step 3: Simplify the Expression

Combine the like terms. The final simplified expression will be: \( u^{3} - u^{2} - u - 5 \)

Key Concepts

Polynomial Arithmetic

Polynomials are algebraic expressions consisting of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. For instance, in the expression \(u^3 - u^2 - u - 5\), each term like \(u^3\), \(u^2\), \(u\), and \(5\) is a polynomial in its own right, and when combined, they form a polynomial equation.

Subtracting polynomials, as shown in our example, is a form of polynomial arithmetic where one polynomial is subtracted from another. It's crucial to follow specific rules, such as distributing the subtraction sign across the terms being subtracted. Additionally, in case of complicated expressions, it is often necessary to rearrange the terms in a way that brings like terms together, which sets the stage for simplifying the expression.

Understanding polynomial arithmetic is important for advancing in algebra, calculus, and various applications in science, economics, and engineering. It informs problem-solving techniques across a wide range of mathematical problems.

Simplifying Expressions

Simplifying an expression means to make it as elementary or as reduced as possible without changing its value. The goal is to have the least amount of terms and the simplest form of those terms. This involves combining like terms, reducing fractions, and applying various algebraic rules to condense the expression.

Steps to Simplify

• Combine like terms: Terms that have the same variable part, such as \(u^3\) and \(u^2\), can be grouped together.
• Apply arithmetic: Perform any addition or subtraction of the coefficients of these like terms.
• Reduce fractions: If the expression includes fractions, they should be reduced to their simplest form when possible.

When simplifying expressions, it's important to carry out operations in the correct order: parentheses first, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). The acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is often used to help remember this sequence.

Combining like terms

Combining like terms is a fundamental process in algebra which simplifies algebraic expressions to make them easier to work with. Like terms are terms that have the same variables raised to the same power. For example, in the terms \(3x^2\) and \(5x^2\), the variable parts \(x^2\) are identical, which means they can be combined.

In the subtraction of polynomials, such as \(u^3 - u - u^2 - 5\), we look for terms that have the same variable to the same exponent:

• Here, there is only one term with \(u^3\), so it remains as is.
• The terms with \(u\) and without variables (the constant term) do not have like terms, so they also remain unchanged.
• The only pair of like terms is \( -u\) and \( -u^2\) which are already arranged next to each other, even though they cannot be combined as their exponents differ.

Combining like terms reduces the complexity of calculations and helps in finding the simplest form of an algebraic expression. It is an essential skill for solving equations, as it allows one to isolate variables and solve for unknowns.