Problem 26 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\left(6 m^{4}-3 m^{2}+m\right)-\left(2 m^{3}+5 m^{2}+4 m\right)+\left(m^{2}-m\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(6m^4 - 2m^3 - 7m^2 - 4m\).

Step 1: Distribute the Negative Sign

The expression given is \( (6m^4-3m^2+m) - (2m^3+5m^2+4m) + (m^2-m) \). Start by distributing the negative sign across the second set of parentheses. This means multiplying each term inside the parentheses by -1:\((6m^4 - 3m^2 + m) - 2m^3 - 5m^2 - 4m + (m^2 - m)\).

Step 2: Combine All Like Terms

Group the expression by terms that have the same powers of \(m\):- \(6m^4\) remains as is since there are no other \(m^4\) terms.- Combine the \(m^3\) terms: \(-2m^3\).- Combine the \(m^2\) terms: \(-3m^2 - 5m^2 + m^2 = -7m^2\).- Combine the \(m\) terms: \(m - 4m - m = -4m\).Now, the resulting expression is: \(6m^4 - 2m^3 - 7m^2 - 4m\).

Step 3: Present the Final Simplified Expression

From Step 2, the simplified expression combining all like terms is \(6m^4 - 2m^3 - 7m^2 - 4m\). This is the sum of the original polynomial expressions.

Key Concepts

Polynomial Simplification

Polynomial simplification is all about making complex polynomial expressions easier to work with by eliminating unwanted complexity. In this process, we aim to rewrite polynomials in their most straightforward form. Simplifying polynomials typically involves:

• Distributing signs across grouped terms
• Combining similar terms
• Ensuring that each term is expressed as clearly as possible

To simplify the expression \((6m^4-3m^2+m) - (2m^3+5m^2+4m) + (m^2-m)\), the first step is dealing with any negative signs impacting the terms inside the parentheses. This makes expression handling cleaner and enables us to focus on combining like terms effectively. Simplification does not change the overall value of the expression; it just makes it less cumbersome to evaluate and manipulate, especially in further calculations or algebraic operations.

Combining Like Terms

Combining like terms is an essential step in polynomial operations. It involves grouping and adding together the polynomial terms that have the same variable and exponent. This action is vital because it helps reduce a polynomial to its simplest possible form. In the given expression, the like terms were managed in groups:

• The only term with \(m^4\) is \(6m^4\).
• For terms in \(m^3\), there was only one present: \(-2m^3\).
• Terms in \(m^2\) involved combining: \(-3m^2\), \(-5m^2\), and \(m^2\), which simplified to \(-7m^2\).
• Linear terms in \(m\) were added as: \(m\), \(-4m\), and \(-m\), leading to \(-4m\).

Through this process, we consolidated the terms and found the most reduced and clear form of the polynomial: \(6m^4 - 2m^3 - 7m^2 - 4m\). It becomes much easier to understand and use once like terms are combined.

Distributive Property

The distributive property is a fundamental element in mathematics and polynomial operations that allows us to break down expressions. It states that multiplying a term by a group of terms inside a parenthesis is equivalent to multiplying each term separately by the outside multiplier. It can be summarized as:\[ a(b + c) = ab + ac \]In our current expression, the distributive property was critically used to handle the negative sign in front of the second set of parentheses. By distributing the negative sign, we effectively negated each term within:

• \(- (2m^3 + 5m^2 + 4m)\) becomes \(-2m^3 - 5m^2 - 4m\).

Using the distributive property here simplifies the arithmetic, cleaves off nuisance parentheses, and paves the way for combining like terms hassle-free. This property often provides foundational support throughout algebra, letting us methodically unfold expressions and solve equations efficiently.