Add the polynomials: [tex]x^2+2 x+1[/tex] and [tex]y^2+2 x+3[/tex]
Answer (1)
To add the polynomials x 2 + 2 x + 1 and y 2 + 2 x + 3 , we combine like terms:
Combine the x terms: 2 x + 2 x = 4 x .
Combine the constant terms: 1 + 3 = 4 .
Write the sum of the polynomials: x 2 + 4 x + 4 + y 2 .
The final answer is x 2 + 4 x + 4 + y 2 .
Explanation
Understanding the Problem We are given two polynomials: x 2 + 2 x + 1 and y 2 + 2 x + 3 . Our goal is to add these two polynomials together.
Adding the Polynomials To add the polynomials, we combine like terms. This means we add the coefficients of the terms with the same variable and exponent.
Combining Like Terms Adding the polynomials:
( x 2 + 2 x + 1 ) + ( y 2 + 2 x + 3 )
= x 2 + 2 x + 1 + y 2 + 2 x + 3
= x 2 + ( 2 x + 2 x ) + y 2 + ( 1 + 3 )
= x 2 + 4 x + y 2 + 4
Finding the Correct Option The sum of the two polynomials is x 2 + 4 x + y 2 + 4 . Now we compare this result with the given options to find the correct answer. The correct answer is x 2 + 4 x + 4 + y 2 .
Examples
Polynomial addition is a fundamental concept in algebra and is used in various real-world applications. For example, when calculating the total cost of items with varying prices and quantities, polynomial addition can be used to combine the individual costs into a single expression. Similarly, in physics, when analyzing the motion of objects, polynomial addition can be used to combine the distances traveled by an object in different time intervals to find the total distance traveled.
