Problem 72 - Algebra A Combined Function

Question

$$ \left[\left(7.9 y^{4}-6.8 y^{3}+3.3 y\right)+\left(6.1 y^{3}-5\right)\right]-\left(4.2 y^{4}+1.1 y-1\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 3.7y^4 - 0.7y^3 + 2.2y - 4 \).

Step 1: Simplify Inside the Brackets

First, simplify each expression inside the brackets. Start with \( (7.9y^4 - 6.8y^3 + 3.3y) + (6.1y^3 - 5) \). Combine like terms to get \( 7.9y^4 - 0.7y^3 + 3.3y - 5 \).

Step 2: Subtract the Right Expression

Now, subtract the expression \( (4.2y^4 + 1.1y - 1) \) from the result of Step 1. Rewrite it as \( 7.9y^4 - 0.7y^3 + 3.3y - 5 - (4.2y^4 + 1.1y - 1) \).

Step 3: Distribute the Negative Sign

Distribute the negative sign across \(4.2y^4 + 1.1y - 1\), resulting in \(-4.2y^4 - 1.1y + 1\). Combine this with the expression from Step 2 to get \( 7.9y^4 - 0.7y^3 + 3.3y - 5 - 4.2y^4 - 1.1y + 1 \).

Step 4: Combine Like Terms

Finally, combine like terms: \( (7.9y^4 - 4.2y^4) + (-0.7y^3) + (3.3y - 1.1y) + (-5 + 1) \). This simplifies to \( 3.7y^4 - 0.7y^3 + 2.2y - 4 \).

Key Concepts

Combining Like Terms

When working with polynomials, one of the most essential skills is combining like terms. This means grouping together terms in an expression that have identical variable parts. These are terms where both the variable and the exponent match. For instance, in the expression \(7.9y^4 - 6.8y^3 + 3.3y\) and \(6.1y^3 - 5\), the terms \(-6.8y^3\) and \(6.1y^3\) are like terms because they both have \(y^3\).
Combining them involves adding or subtracting their coefficients, the numerical part of the terms. In our example, we'll combine \(-6.8 + 6.1\) to get \(-0.7y^3\).

• Identify like terms.
• Only combine terms with exactly the same variable and exponent.
• Add or subtract the coefficients of these terms.

This simplification is key to making polynomials easier to work with, reducing complexity in both calculations and solutions.

Distributive Property

The distributive property is a fundamental algebraic concept frequently used in polynomial simplification. It allows us to multiply a single term by every term inside a parenthesis. This property is particularly handy when an expression includes subtraction within a bracket.
For the expression \(7.9y^4 - 0.7y^3 + 3.3y - 5 - (4.2y^4 + 1.1y - 1)\), we apply the distributive property by distributing the negative sign across the terms inside the bracket:

• Change the sign of each term inside the bracket.
• \(-4.2y^4\) becomes \(-4.2y^4\), \(+1.1y\) becomes \(-1.1y\), \(-1\) becomes \(+1\).

Including the distributive step of changing the signs ensures the polynomial is correctly simplified, avoiding common errors like omitting the change in sign for each term.

Polynomial Subtraction

Subtracting polynomials can seem complex, but it's straightforward with clear steps. It involves subtracting corresponding terms from another polynomial, much like subtraction with numbers, but here each term's sign must be considered.
In the exercise, we are tasked with subtracting \((4.2y^4 + 1.1y - 1)\) from another polynomial. To do this, distribute the negative sign, resulting in \(-4.2y^4 - 1.1y + 1\), then progress by combining these terms with those in the main polynomial.
This calculation becomes:

• Subtract identical terms by carrying over the sign change from the distribution step.
• Simplify by combining these restructured terms together, aligning with the correct signs.

The result is a simplified polynomial where subtraction and combination of like terms are accomplished successfully, leading to the final simplified polynomial \(3.7y^4 - 0.7y^3 + 2.2y - 4\). Understanding and applying these steps in sequence helps demystify polynomial subtraction.