What is the sum of the polynomials?
$\left(8 x^2-9 y^2-4 x\right)+\left(x^2-3 y^2-7 x\right)$
Answer (1)
Combine like terms by adding the coefficients of terms with the same variable and exponent.
Add the x 2 terms: 8 x 2 + x 2 = 9 x 2 .
Add the y 2 terms: − 9 y 2 − 3 y 2 = − 12 y 2 .
Add the x terms: − 4 x − 7 x = − 11 x . The sum of the polynomials is 9 x 2 − 12 y 2 − 11 x .
Explanation
Understanding the problem We are given two polynomials: ( 8 x 2 − 9 y 2 − 4 x ) and ( x 2 − 3 y 2 − 7 x ) . Our goal is to find their sum.
Adding the polynomials To find the sum of the polynomials, we combine like terms. This means we add the coefficients of the terms with the same variables and exponents.
Writing the expression We have:
( 8 x 2 − 9 y 2 − 4 x ) + ( x 2 − 3 y 2 − 7 x )
Grouping like terms Now, let's group the like terms together:
( 8 x 2 + x 2 ) + ( − 9 y 2 − 3 y 2 ) + ( − 4 x − 7 x )
Adding the coefficients Next, we add the coefficients of each group of like terms:
8 x 2 + x 2 = ( 8 + 1 ) x 2 = 9 x 2
− 9 y 2 − 3 y 2 = ( − 9 − 3 ) y 2 = − 12 y 2
− 4 x − 7 x = ( − 4 − 7 ) x = − 11 x
Resulting polynomial So, the sum of the polynomials is:
9 x 2 − 12 y 2 − 11 x
Finding the correct answer Comparing our result with the given options, we find that the correct answer is 9 x 2 − 12 y 2 − 11 x .
Examples
Polynomials are used in various fields such as physics, engineering, and economics. For example, in physics, polynomials can be used to describe the trajectory of a projectile. In economics, they can be used to model cost and revenue functions. Understanding how to add polynomials is a fundamental skill that can be applied in many real-world situations.
