Find the sum: $(x^2+2 x+3)+(3 x^2+x+1)$
A. $4 x^2+3 x+4$
B. $4 x^4+4 x+3$
C. $4 x^3+3 x+4$
D. $3 x^4+4 x+3$
Answer (1)
Add the coefficients of the x 2 terms: 1 x 2 + 3 x 2 = 4 x 2 .
Add the coefficients of the x terms: 2 x + x = 3 x .
Add the constant terms: 3 + 1 = 4 .
The sum of the polynomials is 4 x 2 + 3 x + 4 .
Explanation
Understanding the Problem We are given two polynomials: ( x 2 + 2 x + 3 ) and ( 3 x 2 + x + 1 ) . Our goal is to find their sum.
Combining Like Terms To find the sum of the two polynomials, we need to combine like terms. This means adding the coefficients of the terms with the same power of x .
Grouping Like Terms We have:
( x 2 + 2 x + 3 ) + ( 3 x 2 + x + 1 )
Group the like terms:
( x 2 + 3 x 2 ) + ( 2 x + x ) + ( 3 + 1 )
Adding Coefficients Now, add the coefficients of the like terms:
1 x 2 + 3 x 2 = 4 x 2 2 x + 1 x = 3 x 3 + 1 = 4
So, the sum is 4 x 2 + 3 x + 4 .
Final Result Therefore, the sum of the two polynomials is 4 x 2 + 3 x + 4 .
Examples
Polynomial addition is a fundamental concept in algebra and is used in various real-world applications. For example, in physics, when calculating the total distance traveled by an object with changing velocity, you might add polynomials representing different segments of the journey. Similarly, in economics, polynomial addition can be used to model total cost or revenue functions by combining different cost or revenue components. Understanding polynomial addition helps in simplifying complex expressions and solving practical problems across various disciplines.
