Problem 71 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \text { Subtract }\left(-y^{5}+5 y^{4}-1.2\right) \text { from }\left(2 y^{5}-y^{4}\right) $$

Step-by-Step Solution

Answer

The result is \(3y^5 - 6y^4 + 1.2\).

Step 1: Identify the Expressions

We are given the expressions to work with: the first expression is \(2y^5 - y^4\), and the second one is \(-y^5 + 5y^4 - 1.2\). We need to subtract the second expression from the first.

Step 2: Change Signs in the Second Expression

To subtract \(-y^5 + 5y^4 - 1.2\), change the sign of each term in the expression to get \( y^5 - 5y^4 + 1.2 \).

Step 3: Combine the Expressions

Now combine the expressions by adding the modified second expression to the first expression: \[ (2y^5 - y^4) + (y^5 - 5y^4 + 1.2) \]

Step 4: Simplify the Expression

Combine like terms:* For \(y^5\) terms: \(2y^5 + y^5 = 3y^5\)* For \(y^4\) terms: \(-y^4 - 5y^4 = -6y^4\)* Constant: There is no constant in the first expression, so keep \(1.2\).The simplified expression is:\[ 3y^5 - 6y^4 + 1.2 \]

Step 5: Write the Final Answer

The result of the subtraction is the simplified expression:\[ 3y^5 - 6y^4 + 1.2 \]

Key Concepts

Subtraction of Polynomials

In mathematics, polynomial subtraction involves taking away one polynomial from another. To do this correctly, we need to be careful with the rules of arithmetic, particularly dealing with negative numbers and subtraction. When you see a subtraction problem, the key is to change the problem into an addition problem with opposite signs. This is because subtracting something is the same as adding its opposite.
For example, to subtract the polynomial \(-y^5 + 5y^4 - 1.2 \) from another, \(2y^5 - y^4 \), you first change the signs in the polynomial you are subtracting. In this case, you convert \(-y^5 + 5y^4 - 1.2 \) into \(y^5 - 5y^4 + 1.2 \). This step ensures you're accurately performing the subtraction.

Like Terms

In algebra, like terms are terms in an expression that have the same variable raised to the same power. Recognizing and combining like terms is essential for simplifying polynomials.

• Terms like \(2y^5 \) and \(y^5 \) are considered like terms because they contain the same variable \(y\) raised to the same power, which is 5 in this case.
• On the other hand, \(-y^4\) and \(-5y^4\) are also like terms because they share the variable \(y\) raised to the power of 4.

While combining these terms, simply add or subtract the coefficients (the numbers in front of the variables) to get a single term that represents them both. This method greatly simplifies expressions and is a key step in many algebraic operations.

Simplifying Expressions

Simplifying expressions involves combining like terms to reduce polynomials to their simplest form. When expressions are simplified, they're often easier to understand and work with.
The expression \((2y^5 - y^4) + (y^5 - 5y^4 + 1.2) \) can be simplified by combining like terms:

• Combine the \(y^5\) terms: \(2y^5\) and \(y^5\) to get \(3y^5\).
• Combine the \(y^4\) terms: \(-y^4\) and \(-5y^4\) to get \(-6y^4\).
• Finally, add any constant values. In this exercise, the only constant is \(1.2\).

As a result, the expression simplifies to \(3y^5 - 6y^4 + 1.2\). This process makes the polynomial much more manageable and prepares it for potential further operations or interpretations.