Find the sum of these polynomials.
$\left(x^2-x+7\right)+\left(9 x^2+6\right)=$
A. $10 x^2-x+13$
B. $10 x^2-x+1$
C. $9 x^2-x+1$
D. $9 x^2-x+13$
Answer (1)
Combine like terms: Add the coefficients of the x 2 terms, x terms, and constant terms.
Add x 2 terms: x 2 + 9 x 2 = 10 x 2 .
Add constant terms: 7 + 6 = 13 .
The sum of the polynomials is 10 x 2 − x + 13 .
Explanation
Understanding the Problem We are given two polynomials, ( x 2 − x + 7 ) and ( 9 x 2 + 6 ) , and we need to find their sum. This involves combining like terms.
Combining Like Terms To add the polynomials, we combine the terms with the same power of x . That is, we add the x 2 terms together, the x terms together, and the constant terms together.
( x 2 − x + 7 ) + ( 9 x 2 + 6 ) = ( x 2 + 9 x 2 ) + ( − x ) + ( 7 + 6 )
Simplifying the Expression Now, we simplify the expression:
( x 2 + 9 x 2 ) = 10 x 2 ( − x ) = − x ( 7 + 6 ) = 13
So, the sum of the polynomials is:
10 x 2 − x + 13
Final Answer The sum of the polynomials ( x 2 − x + 7 ) and ( 9 x 2 + 6 ) is 10 x 2 − x + 13 . Therefore, the correct answer is A.
Examples
Polynomials are used in many real-world applications, such as modeling curves and shapes in engineering and computer graphics. For example, engineers might use polynomials to design the shape of a bridge or the trajectory of a rocket. Computer graphics designers use polynomials to create smooth curves and surfaces for objects in video games and movies. By adding polynomials, we can combine different models or shapes to create more complex designs. For instance, if we have a polynomial representing the shape of a car body and another representing the shape of a spoiler, we can add them to see the combined shape.
