Given the following functions:
[tex]
\begin{array}{l}
f(x)=3 x^2+5 x-2 \\
g(x)=5 x^3-4 x^2+4
\end{array}
[/tex]
Find [tex]$(f+g)(x)$[/tex]
A. [tex]$(f+y)(x)=5 x^3-x^2+5 x+2$[/tex]
B. [tex]$(f+g)(x)=8 x^3+x+2$[/tex]
C. [tex]$(f+g)(x)=-5 x^3+7 x^2+5 x-6$[/tex]
D. [tex]$(f+g)(x)=5 x^3+3 x^2+5 x+2$[/tex]
Answer (2)
Add the two polynomials: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the expressions: ( f + g ) ( x ) = ( 3 x 2 + 5 x − 2 ) + ( 5 x 3 − 4 x 2 + 4 ) .
Combine like terms: ( f + g ) ( x ) = 5 x 3 + ( 3 x 2 − 4 x 2 ) + 5 x + ( − 2 + 4 ) .
Simplify to get the final answer: ( f + g ) ( x ) = 5 x 3 − x 2 + 5 x + 2 . The answer is 5 x 3 − x 2 + 5 x + 2 .
Explanation
Understanding the Problem We are given two polynomial functions:
f ( x ) = 3 x 2 + 5 x − 2
g ( x ) = 5 x 3 − 4 x 2 + 4
We need to find the sum of these two functions, which is denoted as ( f + g ) ( x ) . This means we need to add the two polynomials together.
Adding the Polynomials To find ( f + g ) ( x ) , we simply add the two polynomials f ( x ) and g ( x ) :
( f + g ) ( x ) = f ( x ) + g ( x )
Substitute the given expressions for f ( x ) and g ( x ) :
( f + g ) ( x ) = ( 3 x 2 + 5 x − 2 ) + ( 5 x 3 − 4 x 2 + 4 )
Combining Like Terms Now, we combine like terms. This means we group together terms with the same power of x :
( f + g ) ( x ) = 5 x 3 + ( 3 x 2 − 4 x 2 ) + 5 x + ( − 2 + 4 )
Simplify the expression:
( f + g ) ( x ) = 5 x 3 − x 2 + 5 x + 2
Final Answer The resulting polynomial is ( f + g ) ( x ) = 5 x 3 − x 2 + 5 x + 2 . Comparing this to the given options, we see that it matches option A.
Therefore, the correct answer is A. ( f + g ) ( x ) = 5 x 3 − x 2 + 5 x + 2
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 aerodynamic profile of an airplane wing. In economics, polynomials can be used to represent cost and revenue functions, helping businesses analyze their profitability. Computer graphics also rely heavily on polynomials to create smooth and realistic images.
We find that ( f + g ) ( x ) = 5 x 3 − x 2 + 5 x + 2 by adding the polynomials and combining like terms. This matches option A in the multiple choice answers provided. Thus, the correct option is A.
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