Subtract: \left(-8 Y^5 + 4\right) - \left(13 Y^5 - 6\right ]Your Answer Should Be In Simplest Terms.

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Understanding the Problem

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When subtracting polynomials, we need to combine like terms and simplify the expression. In this problem, we are given two polynomials: $\left(-8 y^5 + 4\right)$ and $\left(13 y^5 - 6\right)$. Our goal is to subtract the second polynomial from the first one and simplify the result.

Step 1: Distribute the Negative Sign

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To subtract the second polynomial from the first one, we need to distribute the negative sign to all the terms in the second polynomial. This means that we will change the sign of each term in the second polynomial.

\left(-8 y^5 + 4\right) - \left(13 y^5 - 6\right) = -8 y^5 + 4 - 13 y^5 + 6

Step 2: Combine Like Terms

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Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have two terms with the variable $y^5$ and two constant terms.

-8 y^5 + 4 - 13 y^5 + 6 = -8 y^5 - 13 y^5 + 4 + 6

Step 3: Simplify the Expression

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Now that we have combined like terms, we can simplify the expression by combining the coefficients of the like terms.

-8 y^5 - 13 y^5 + 4 + 6 = -21 y^5 + 10

Step 4: Write the Final Answer

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The final answer is $\boxed{-21 y^5 + 10}$.

Conclusion

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Subtracting polynomials involves combining like terms and simplifying the expression. By following the steps outlined above, we can simplify the given expression and write the final answer.

Tips and Tricks

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• When subtracting polynomials, make sure to distribute the negative sign to all the terms in the second polynomial.
• Combine like terms by adding or subtracting the coefficients of the like terms.
• Simplify the expression by combining the coefficients of the like terms.

Real-World Applications

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Subtracting polynomials has many real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, subtracting polynomials can be used to calculate the stress on a beam or the force on a spring. In physics, subtracting polynomials can be used to calculate the energy of a system or the momentum of an object. In computer science, subtracting polynomials can be used to optimize algorithms and improve the performance of computer programs.

Common Mistakes

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• Failing to distribute the negative sign to all the terms in the second polynomial.
• Failing to combine like terms.
• Failing to simplify the expression.

Practice Problems

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• Subtract the following polynomials: $\left(2 x^3 - 3 x^2 + 4 x - 1\right)$ and $\left(-2 x^3 + 3 x^2 - 4 x + 1\right)$.
• Subtract the following polynomials: $\left(3 y^4 + 2 y^3 - 4 y^2 + 1\right)$ and $\left(-3 y^4 - 2 y^3 + 4 y^2 - 1\right)$.

Solutions

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• $\left(2 x^3 - 3 x^2 + 4 x - 1\right) - \left(-2 x^3 + 3 x^2 - 4 x + 1\right) = 4 x^3 - 6 x^2 + 8 x - 2$
• $\left(3 y^4 + 2 y^3 - 4 y^2 + 1\right) - \left(-3 y^4 - 2 y^3 + 4 y^2 - 1\right) = 6 y^4 + 4 y^3 - 8 y^2 + 2$
  

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Frequently Asked Questions

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Q: What is the first step in subtracting polynomials?

A: The first step in subtracting polynomials is to distribute the negative sign to all the terms in the second polynomial.

Q: How do I combine like terms when subtracting polynomials?

A: To combine like terms when subtracting polynomials, add or subtract the coefficients of the like terms.

Q: What is the final step in subtracting polynomials?

A: The final step in subtracting polynomials is to simplify the expression by combining the coefficients of the like terms.

Q: What are some common mistakes to avoid when subtracting polynomials?

A: Some common mistakes to avoid when subtracting polynomials include failing to distribute the negative sign to all the terms in the second polynomial, failing to combine like terms, and failing to simplify the expression.

Q: How do I know if I have combined like terms correctly?

A: To check if you have combined like terms correctly, make sure that you have added or subtracted the coefficients of the like terms and that you have simplified the expression.

Q: Can I use a calculator to subtract polynomials?

A: Yes, you can use a calculator to subtract polynomials. However, it is also important to understand the steps involved in subtracting polynomials and to be able to do it by hand.

Q: How do I apply the concept of subtracting polynomials to real-world problems?

A: The concept of subtracting polynomials can be applied to real-world problems in fields such as engineering, physics, and computer science. For example, in engineering, subtracting polynomials can be used to calculate the stress on a beam or the force on a spring.

Advanced Questions

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Q: How do I subtract polynomials with variables of different degrees?

A: To subtract polynomials with variables of different degrees, follow the same steps as before, but make sure to combine like terms carefully.

Q: How do I subtract polynomials with variables of the same degree but different coefficients?

A: To subtract polynomials with variables of the same degree but different coefficients, combine the coefficients of the like terms and simplify the expression.

Q: How do I subtract polynomials with variables of the same degree and the same coefficient but different signs?

A: To subtract polynomials with variables of the same degree and the same coefficient but different signs, combine the coefficients of the like terms and simplify the expression.

Example Problems

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Problem 1

Subtract the following polynomials: $\left(2 x^3 - 3 x^2 + 4 x - 1\right)$ and $\left(-2 x^3 + 3 x^2 - 4 x + 1\right)$.

Solution

$\left(2 x^3 - 3 x^2 + 4 x - 1\right) - \left(-2 x^3 + 3 x^2 - 4 x + 1\right) = 4 x^3 - 6 x^2 + 8 x - 2$

Problem 2

Subtract the following polynomials: $\left(3 y^4 + 2 y^3 - 4 y^2 + 1\right)$ and $\left(-3 y^4 - 2 y^3 + 4 y^2 - 1\right)$.

Solution

$\left(3 y^4 + 2 y^3 - 4 y^2 + 1\right) - \left(-3 y^4 - 2 y^3 + 4 y^2 - 1\right) = 6 y^4 + 4 y^3 - 8 y^2 + 2$

Practice Problems

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• Subtract the following polynomials: $\left(2 x^3 - 3 x^2 + 4 x - 1\right)$ and $\left(-2 x^3 + 3 x^2 - 4 x + 1\right)$.
• Subtract the following polynomials: $\left(3 y^4 + 2 y^3 - 4 y^2 + 1\right)$ and $\left(-3 y^4 - 2 y^3 + 4 y^2 - 1\right)$.

Solutions

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• $\left(2 x^3 - 3 x^2 + 4 x - 1\right) - \left(-2 x^3 + 3 x^2 - 4 x + 1\right) = 4 x^3 - 6 x^2 + 8 x - 2$
• $\left(3 y^4 + 2 y^3 - 4 y^2 + 1\right) - \left(-3 y^4 - 2 y^3 + 4 y^2 - 1\right) = 6 y^4 + 4 y^3 - 8 y^2 + 2$