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Chapter 10: Problem 508

Add or subtract the polynomials. $$\left(y^{2}-3 y+12\right)+\left(5 y^{2}-9\right)$$

Short Answer

Expert verified
6y^2 - 3y + 3

Step by step solution

01

Identify Like Terms

Match the terms from each polynomial that have the same degree. In the given polynomials, identify like terms as follows: From the first polynomial: y^2, -3y, 12 From the second polynomial: 5y^2, -9
02

Combine Like Terms

Add the coefficients of the like terms together: For the quadratic terms: y^2 + 5y^2 = 6y^2 For the linear terms: -3y (no corresponding term from the second polynomial, so it remains -3y) For the constant terms: 12 + (-9) = 3
03

Write the Resulting Polynomial

Combine the results from the addition of coefficients to form the final polynomial: 6y^2 - 3y + 3

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Key Concepts

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

Combining Like Terms
Combining like terms is a fundamental concept in algebra that simplifies expressions by merging terms with the same variables and exponents. In our exercise, we identified like terms from two polynomials. Terms like \(y^2\) and \(5y^2\), which have the same variable raised to the same power, are combined by adding their coefficients.
Consider these steps:
  • Match terms with the same variable and exponent.
  • Add or subtract their coefficients.
For instance, when we had \(y^2 + 5y^2\), we added the coefficients (1 + 5) to get \(6y^2\). This process simplified the polynomial and made it easier to interpret and use it in subsequent calculations. Understanding this principle is vital for more advanced algebraic manipulations.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2, meaning its highest exponent on the variable is 2. The general form is \(ax^2 + bx + c\).
In our exercise, we worked with the quadratic polynomial \(y^2 - 3y + 12\).
The specific characteristics of quadratic polynomials include:
  • The presence of a quadratic term like \(y^2\).
  • A linear term (if it exists), like \(-3y\).
  • A constant term, such as 12.
Adding or subtracting quadratic polynomials involves combining like terms, focusing particularly on the terms with \(y^2\), as shown in the exercise. It's essential to keep track of each term鈥檚 degree to perform the operations correctly. Quadratic polynomials frequently appear in various mathematical contexts, including physics and engineering, so mastering these concepts early is important.
Prealgebra
Prealgebra forms the foundation for more advanced mathematical studies. It includes fundamental topics like basic operations, understanding variables, and solving simple equations.
In the context of our exercise, prealgebra skills are seen through:
  • Recognizing and identifying like terms.
  • Performing basic addition and subtraction of polynomials.
For example, knowing that \(y^2\) terms are combined separately from \(-3y\) or a constant like 12, forms a crucial part of algebra readiness. Combining terms in \( y^2 - 3y + 12 + 5y^2 - 9 \) requires these foundational skills. Proper grasp of prealgebra ensures a smoother transition to more complex algebraic techniques, paving the way for success in higher mathematics.

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