Lorne subtracted $6 x^3-2 x+3$ from $-3 x^3+5 x^2+4 x-7$. Use the drop-down menus to identify the steps Lorne used to find the difference.
1. $\left(-3 x^3+5 x^2+4 x-7\right)+\left(-6 x^3+2 x-3\right)$
2. $\left(-3 x^3\right)+5 x^2+4 x+(-7)+\left(-6 x^3\right)+2 x+(-3)$
3. $\left[\left(-3 x^3\right)+\left(-6 x^3\right)\right]+[4 x+2 x]+[(-7)+(-3)]+\left[5 x^2\right]$
4. $-9 x^3+6 x+(-10)+5 x^2$
5. $-9 x^3+5 x^2+6 x-10
Answer (2)
Lorne's process involves distributing the negative sign for subtraction, removing parentheses to combine terms, grouping like terms together, and then simplifying to reach the final polynomial in standard form. The selected steps are captured in options 1 through 5. Therefore, the accurate sequence of Lorne's calculations matches these provided steps.
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Distribute the negative sign: ( − 3 x 3 + 5 x 2 + 4 x − 7 ) − ( 6 x 3 − 2 x + 3 ) = ( − 3 x 3 + 5 x 2 + 4 x − 7 ) + ( − 6 x 3 + 2 x − 3 ) .
Remove the parentheses: − 3 x 3 + 5 x 2 + 4 x − 7 − 6 x 3 + 2 x − 3 = ( − 3 x 3 ) + 5 x 2 + 4 x + ( − 7 ) + ( − 6 x 3 ) + 2 x + ( − 3 ) .
Group like terms: ( − 3 x 3 − 6 x 3 ) + ( 5 x 2 ) + ( 4 x + 2 x ) + ( − 7 − 3 ) = [( − 3 x 3 ) + ( − 6 x 3 )] + [ 4 x + 2 x ] + [( − 7 ) + ( − 3 )] + [ 5 x 2 ] .
Combine like terms and write in standard form: − 9 x 3 + 5 x 2 + 6 x − 10 = − 9 x 3 + 5 x 2 + 6 x − 10 .
Explanation
Understanding the Problem We are given the problem of subtracting the polynomial 6 x 3 − 2 x + 3 from the polynomial − 3 x 3 + 5 x 2 + 4 x − 7 . We need to identify the correct steps to perform this subtraction.
Distributing the Negative Sign Step 1: Distribute the negative sign to the polynomial being subtracted. This is equivalent to adding the negative of the polynomial: ( − 3 x 3 + 5 x 2 + 4 x − 7 ) − ( 6 x 3 − 2 x + 3 ) = ( − 3 x 3 + 5 x 2 + 4 x − 7 ) + ( − 6 x 3 + 2 x − 3 ) . This matches option 1.
Removing Parentheses Step 2: Remove the parentheses: − 3 x 3 + 5 x 2 + 4 x − 7 − 6 x 3 + 2 x − 3 = ( − 3 x 3 ) + 5 x 2 + 4 x + ( − 7 ) + ( − 6 x 3 ) + 2 x + ( − 3 ) . This matches option 2.
Grouping Like Terms Step 3: Group like terms: ( − 3 x 3 − 6 x 3 ) + ( 5 x 2 ) + ( 4 x + 2 x ) + ( − 7 − 3 ) = [( − 3 x 3 ) + ( − 6 x 3 )] + [ 4 x + 2 x ] + [( − 7 ) + ( − 3 )] + [ 5 x 2 ] . This matches option 3.
Combining Like Terms Step 4: Combine like terms: − 9 x 3 + 5 x 2 + 6 x − 10 = − 9 x 3 + 6 x + ( − 10 ) + 5 x 2 . This matches option 4.
Final Result Step 5: Write the polynomial in standard form (decreasing order of exponents): − 9 x 3 + 5 x 2 + 6 x − 10 = − 9 x 3 + 5 x 2 + 6 x − 10 . This matches option 5. Therefore, the steps Lorne used are exactly the steps provided.
Examples
Polynomial subtraction is a fundamental concept in algebra and has numerous real-world applications. For instance, consider a scenario where a company's revenue and costs are modeled by polynomial functions. Subtracting the cost polynomial from the revenue polynomial yields the profit polynomial. Analyzing the profit polynomial can help the company make informed decisions about pricing, production levels, and cost management. Understanding polynomial operations is crucial for modeling and optimizing various real-world situations.
