Subtract these polynomials.
$\left(6 x^2-x+8\right)-\left(x^2+2\right)=$
A. $7 x^2-x+10$
B. $5 x^2-x+10$
C. $5 x^2-x+6$
D. $7 x^2-x+6$
Answer (2)
Distribute the negative sign: ( 6 x 2 − x + 8 ) − ( x 2 + 2 ) = 6 x 2 − x + 8 − x 2 − 2 .
Combine like terms: ( 6 x 2 − x 2 ) − x + ( 8 − 2 ) .
Simplify: 5 x 2 − x + 6 .
The final answer is 5 x 2 − x + 6 .
Explanation
Understanding the Problem We are asked to subtract the polynomial ( x 2 + 2 ) from the polynomial ( 6 x 2 − x + 8 ) . This involves distributing the negative sign and combining like terms.
Distributing the Negative Sign First, distribute the negative sign to each term in the second polynomial: ( 6 x 2 − x + 8 ) − ( x 2 + 2 ) = 6 x 2 − x + 8 − x 2 − 2
Combining Like Terms Next, combine like terms. We have 6 x 2 and − x 2 as the x 2 terms, − x as the x term, and 8 and − 2 as the constant terms. Combining these gives: ( 6 x 2 − x 2 ) − x + ( 8 − 2 ) = 5 x 2 − x + 6
Final Answer The resulting polynomial is 5 x 2 − x + 6 . Comparing this to the multiple choice answers, we see that it matches option C.
Examples
Polynomial subtraction is used in various applications, such as calculating the change in profit or cost when production levels change. For example, if a company's revenue is represented by the polynomial 6 x 2 − x + 8 and their cost is represented by the polynomial x 2 + 2 , where x is the number of units produced, then the profit is the difference between the revenue and the cost. Subtracting the cost polynomial from the revenue polynomial gives the profit polynomial 5 x 2 − x + 6 . This allows the company to easily determine their profit for any given production level.
To subtract the polynomials ( 6 x 2 − x + 8 ) − ( x 2 + 2 ) , we first distribute the negative sign and then combine like terms, resulting in 5 x 2 − x + 6 . This matches option C in the multiple-choice answers. Thus, the final answer is 5 x 2 − x + 6 .
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