The variables $A, B$, and $C$ represent polynomials where $A=x+1, B=x^2+2 x-1$, and $C=2 x$. What is $A B+C$ in simplest form?
A. $x^3+3 x-1$
B. $x^3+4 x-1$
C. $x^3+3 x^2+3 x-1$
D. $x^3+2 x^2-x+1$
Answer (1)
Multiply polynomials A and B: ( x + 1 ) ( x 2 + 2 x − 1 ) = x 3 + 3 x 2 + x − 1 .
Add polynomial C to the result: ( x 3 + 3 x 2 + x − 1 ) + 2 x = x 3 + 3 x 2 + 3 x − 1 .
Simplify the expression by combining like terms.
The final answer is: x 3 + 3 x 2 + 3 x − 1 .
Explanation
Understanding the Problem We are given three polynomials: A = x + 1 , B = x 2 + 2 x − 1 , and C = 2 x . Our goal is to find the simplified form of the expression A B + C .
Multiplying Polynomials A and B First, we need to multiply polynomial A by polynomial B . This means we need to calculate ( x + 1 ) ( x 2 + 2 x − 1 ) . We can use the distributive property (also known as the FOIL method) to do this:
( x + 1 ) ( x 2 + 2 x − 1 ) = x ( x 2 + 2 x − 1 ) + 1 ( x 2 + 2 x − 1 )
Distributing x and 1 Now, we distribute x and 1 across the terms in the parentheses:
x ( x 2 + 2 x − 1 ) = x 3 + 2 x 2 − x 1 ( x 2 + 2 x − 1 ) = x 2 + 2 x − 1
So, ( x + 1 ) ( x 2 + 2 x − 1 ) = x 3 + 2 x 2 − x + x 2 + 2 x − 1
Combining Like Terms Next, we combine like terms in the expression x 3 + 2 x 2 − x + x 2 + 2 x − 1 :
x 3 + ( 2 x 2 + x 2 ) + ( − x + 2 x ) − 1 = x 3 + 3 x 2 + x − 1
So, A B = x 3 + 3 x 2 + x − 1
Adding Polynomial C Now we need to add polynomial C to the result we just found. That is, we need to calculate ( x 3 + 3 x 2 + x − 1 ) + ( 2 x ) .
( x 3 + 3 x 2 + x − 1 ) + ( 2 x ) = x 3 + 3 x 2 + ( x + 2 x ) − 1 = x 3 + 3 x 2 + 3 x − 1
Final Result Therefore, A B + C = x 3 + 3 x 2 + 3 x − 1 .
Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design bridges and predict their stability under different loads. Similarly, economists use polynomials to model economic growth and predict future trends. Understanding polynomial operations like multiplication and addition is crucial for these applications.
