Which polynomial represents the sum below?
$\begin{array}{r}
3 x^2+7 x+3 \\
+\quad 5 x^2+12 x \\
\hline
\end{array}$
A. $15 x^2+19 x+3$
B. $8 x^2+84 x+3$
C. $15 x^2+84 x+3$
D. $8 x^2+19 x+3$
Answer (1)
Add the coefficients of the x 2 terms: 3 + 5 = 8 , resulting in 8 x 2 .
Add the coefficients of the x terms: 7 + 12 = 19 , resulting in 19 x .
The constant term is 3.
Combine the terms to get the final polynomial: 8 x 2 + 19 x + 3 .
Explanation
Understanding the Problem We are given two polynomials, 3 x 2 + 7 x + 3 and 5 x 2 + 12 x , and we want to find their sum.
Adding Like Terms To add the polynomials, we combine like terms. This means we add the coefficients of the x 2 terms, the coefficients of the x terms, and the constant terms separately.
Adding x 2 Terms First, let's add the x 2 terms: 3 x 2 + 5 x 2 = ( 3 + 5 ) x 2 = 8 x 2 .
Adding x Terms Next, let's add the x terms: 7 x + 12 x = ( 7 + 12 ) x = 19 x .
Constant Term Finally, the constant term in the first polynomial is 3, and there is no constant term in the second polynomial. So, the constant term in the sum is 3.
Final Sum Therefore, the sum of the two polynomials is 8 x 2 + 19 x + 3 .
Examples
Polynomials are used in various fields like physics, engineering, and economics to model complex relationships. For example, the trajectory of a projectile can be modeled using a quadratic polynomial. By adding polynomials, we can combine different models to create a more comprehensive representation of a system. In finance, polynomials can represent cost functions or revenue streams, and adding them allows for analyzing the overall profitability of a project.
