Which polynomial represents the sum below?
$\left(18 x^2-18\right)+\left(-13 x^2-13 x+13\right)$
A. $18 x^2-31 x-18$
B. $5 x^2-13 x-5$
C. $5 x^2-13 x-31$
D. $31 x^2-31 x-5$
Answer (1)
Combine like terms from the given polynomials: ( 18 x 2 − 18 ) + ( − 13 x 2 − 13 x + 13 ) .
Group the like terms: ( 18 x 2 − 13 x 2 ) + ( − 13 x ) + ( − 18 + 13 ) .
Simplify the expression: 5 x 2 − 13 x − 5 .
The sum of the polynomials is 5 x 2 − 13 x − 5 .
Explanation
Understanding the Problem We are asked to find the sum of two polynomials: ( 18 x 2 − 18 ) and ( − 13 x 2 − 13 x + 13 ) . To do this, we combine like terms.
Combining Like Terms We add the two polynomials by grouping like terms together:
( 18 x 2 − 18 ) + ( − 13 x 2 − 13 x + 13 ) = ( 18 x 2 − 13 x 2 ) + ( − 13 x ) + ( − 18 + 13 )
Simplifying the Expression Now, we simplify the expression by performing the arithmetic operations:
18 x 2 − 13 x 2 = 5 x 2
− 18 + 13 = − 5
So, the sum is 5 x 2 − 13 x − 5 .
Final Answer Therefore, the polynomial that represents the sum is 5 x 2 − 13 x − 5 . Looking at the multiple-choice options, we see that this corresponds to option B.
Examples
Polynomials are used in many areas of mathematics and science. For example, engineers use polynomials to model curves and surfaces, such as the shape of a bridge or the trajectory of a rocket. In economics, polynomials can be used to represent cost and revenue functions, helping businesses make decisions about pricing and production. Understanding how to add and subtract polynomials is a fundamental skill that is used in these and other applications.
