Which polynomial represents the difference below?
$\left(8 x^2+9\right)-\left(3 x^2+2 x+5\right)$
A. $11 x^2+2 x+14$
B. $5 x^2-2 x+4$
C. $5 x^2-2 x+14$
D. $11 x^2-2 x+4$
Answer (1)
Distribute the negative sign: 8 x 2 + 9 − 3 x 2 − 2 x − 5 .
Combine like terms: ( 8 x 2 − 3 x 2 ) − 2 x + ( 9 − 5 ) .
Simplify: 5 x 2 − 2 x + 4 .
The difference is 5 x 2 − 2 x + 4 .
Explanation
Understanding the Problem We are given two polynomials: ( 8 x 2 + 9 ) and ( 3 x 2 + 2 x + 5 ) . We need to find the difference between these two polynomials: ( 8 x 2 + 9 ) − ( 3 x 2 + 2 x + 5 ) .
Strategy To find the difference, we need to subtract the second polynomial from the first. This involves distributing the negative sign to each term in the second polynomial and then combining like terms.
Calculations First, distribute the negative sign to the second polynomial: 8 x 2 + 9 − ( 3 x 2 + 2 x + 5 ) = 8 x 2 + 9 − 3 x 2 − 2 x − 5 Next, combine the like terms: ( 8 x 2 − 3 x 2 ) + ( − 2 x ) + ( 9 − 5 ) Now, simplify the expression: 5 x 2 − 2 x + 4
Final Answer The resulting polynomial is 5 x 2 − 2 x + 4 , which corresponds to option B.
Examples
Polynomial subtraction is a fundamental concept in algebra and is used in various real-life applications. For example, if you are managing a budget and want to determine the remaining funds after deducting expenses, you can represent your initial funds and expenses as polynomials and subtract them to find the remaining balance. Similarly, in physics, you might use polynomial subtraction to find the net force acting on an object when multiple forces are involved.
