Subtract these polynomials.
$\left(2 x^2+4 x+3\right)-\left(4 x^2-2 x-3\right)=$
A. $-2 x^2+6 x+6$
B. $6 x^2+2 x+0$
C. $-2 x^2+2 x+0$
D. $6 x^2+6 x+6$
Answer (1)
Distribute the negative sign: − ( 4 x 2 − 2 x − 3 ) = − 4 x 2 + 2 x + 3 .
Combine like terms: ( 2 x 2 + 4 x + 3 ) + ( − 4 x 2 + 2 x + 3 ) = ( 2 x 2 − 4 x 2 ) + ( 4 x + 2 x ) + ( 3 + 3 ) .
Simplify the expression: − 2 x 2 + 6 x + 6 .
The result of the subtraction is − 2 x 2 + 6 x + 6 .
Explanation
Understanding the Problem We are asked to subtract the polynomial ( 4 x 2 − 2 x − 3 ) from the polynomial ( 2 x 2 + 4 x + 3 ) . This means we need to evaluate the expression ( 2 x 2 + 4 x + 3 ) − ( 4 x 2 − 2 x − 3 ) .
Distributing the Negative Sign To subtract the polynomials, we distribute the negative sign to each term in the second polynomial: − ( 4 x 2 − 2 x − 3 ) = − 4 x 2 + 2 x + 3 .
Combining Like Terms Now, we combine like terms: ( 2 x 2 + 4 x + 3 ) + ( − 4 x 2 + 2 x + 3 ) = ( 2 x 2 − 4 x 2 ) + ( 4 x + 2 x ) + ( 3 + 3 ) .
Simplifying the Expression Simplifying the expression, we get: ( 2 x 2 − 4 x 2 ) + ( 4 x + 2 x ) + ( 3 + 3 ) = − 2 x 2 + 6 x + 6 .
Final Answer Therefore, ( 2 x 2 + 4 x + 3 ) − ( 4 x 2 − 2 x − 3 ) = − 2 x 2 + 6 x + 6 . The correct answer is A.
Examples
Polynomial subtraction is a fundamental concept in algebra and is used in various real-life applications. For instance, consider a scenario where a company's revenue and costs are modeled by polynomials. Subtracting the cost polynomial from the revenue polynomial gives the profit polynomial, which helps in analyzing the company's financial performance. Similarly, in physics, polynomial subtraction can be used to find the net force acting on an object when multiple forces are involved. Understanding polynomial subtraction is crucial for solving problems in engineering, economics, and computer science.
