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Chapter 5: Problem 75

Add. $$ \begin{array}{l}{6 y^{3}-9 y^{2} \quad+8} \\ {4 y^{3}+2 y^{2}+5 y} \\\ \hline\end{array} $$

Short Answer

Expert verified
The answer is \(10y^3 - 7y^2 + 5y + 8\).

Step by step solution

01

- Write down the problem

We need to add the polynomials: \(6y^3 - 9y^2 + 8\) and \(4y^3 + 2y^2 + 5y\).
02

- Group like terms

Identify and group the like terms of the polynomials:- The \(y^3\) terms: \(6y^3\) and \(4y^3\).- The \(y^2\) terms: \(-9y^2\) and \(2y^2\).- The \(y\) terms: \(5y\) (only in second polynomial).- The constant term: \(8\) (only in first polynomial).
03

- Add the like terms

Add the coefficients of the like terms:- For \(y^3\): \(6 + 4 = 10\), so \(10y^3\).- For \(y^2\): \(-9 + 2 = -7\), so \(-7y^2\).- For \(y\): \(0 + 5 = 5\), so \(5y\).- The constant remains the same: \(8\).
04

- Write the final answer

Combine all the terms: \(10y^3 - 7y^2 + 5y + 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

like terms
Understanding 'like terms' is critical when working with polynomials. Like terms are terms that have the same variable raised to the same power. For example, in the problem we looked at, terms like 6y^3 and 4y^3 are considered like terms because they both have the variable y raised to the third power. Similarly, -9y^2 and 2y^2 are like terms because they both have y raised to the second power.
However, terms such as 5y and 8 are not like terms with any other terms in the given polynomials. When adding polynomials, it's essential to group these like terms together to simplify the process. Always remember: ONLY combine like terms.
coefficients
Coefficients play a crucial role in polynomial operations. A coefficient is the number in front of a variable. For instance, in 6y^3, the coefficient is 6, and in -9y^2, it is -9. These numbers tell you how many times to multiply the variable.
In the addition problem we're solving, it's key to focus on combining these coefficients for like terms. For example, for the y^3 terms, we add the coefficients 6 and 4 to get 10. Similarly, for the y^2 terms, we add -9 and 2 to get -7. Understanding coefficients and their role in polynomial operations will make solving these types of problems much more manageable.
combining polynomials
Combining polynomials involves adding or subtracting like terms. The key steps are:
  • Group the like terms.
  • Add or subtract the coefficients.
  • Write the final polynomial.
In our example, we combined two polynomials:
6y^3 - 9y^2 + 8 and 4y^3 + 2y^2 + 5y.
We grouped like terms first, then we added the coefficients of y^3 terms, y^2 terms, and so on. Finally, we rewrote the expression with the new coefficients. Combining polynomials step-by-step ensures you don't miss any terms and that your final answer is simplified correctly.
step-by-step solution
A step-by-step solution is invaluable for understanding the process of polynomial addition. Let's recap the steps for our problem:
  • Step 1: Write down the problem.
  • Step 2: Identify and group like terms: 6y^3 and 4y^3, -9y^2 and 2y^2, and other terms.
  • Step 3: Add the coefficients: 6 + 4 = 10y^3, -9 + 2 = -7y^2, and so forth.
  • Step 4: Write the final answer: 10y^3 - 7y^2 + 5y + 8.
Following these steps methodically ensures accuracy and reinforces your understanding of polynomial addition. Practice will make these steps second nature.

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