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. $-9 x^3+5 x^2+6 x-10$

Lorne performed polynomial subtraction by rewriting it as addition of the additive inverse, then grouped and combined like terms to arrive at the final polynomial − 9 x 3 + 5 x 2 + 6 x − 10 .
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Answered by Anonymous | 2025-07-04

Rewrite the subtraction as addition of the additive inverse.
Write each polynomial as the addition of its terms.
Group like terms together.
Combine like terms and write the polynomial in standard form. The result is − 9 x 3 + 5 x 2 + 6 x − 10 ​ .

Explanation

Understanding the Problem The problem asks us to identify the steps Lorne used to subtract the polynomial 6 x 3 − 2 x + 3 from the polynomial − 3 x 3 + 5 x 2 + 4 x − 7 . This involves understanding how polynomial subtraction works and recognizing the properties applied in each step.

Step 1: Rewriting as Addition Step 1: ( − 3 x 3 + 5 x 2 + 4 x − 7 ) + ( − 6 x 3 + 2 x − 3 ) . This step shows the subtraction being rewritten as the addition of the additive inverse. Subtracting a polynomial is equivalent to adding its opposite.

Step 2: Writing as Addition of Terms Step 2: ( − 3 x 3 ) + 5 x 2 + 4 x + ( − 7 ) + ( − 6 x 3 ) + 2 x + ( − 3 ) . Here, each polynomial is written as the addition of its terms. This makes it easier to rearrange and combine like terms.

Step 3: Grouping Like Terms Step 3: [ ( − 3 x 3 ) + ( − 6 x 3 ) ] + [ 4 x + 2 x ] + [( − 7 ) + ( − 3 )] + [ 5 x 2 ] . In this step, like terms are grouped together. This prepares for the combination of these terms in the next step.

Step 4: Combining Like Terms Step 4: − 9 x 3 + 6 x + ( − 10 ) + 5 x 2 . This step combines the like terms that were grouped in the previous step. For example, − 3 x 3 and − 6 x 3 are combined to give − 9 x 3 , 4 x and 2 x are combined to give 6 x , and − 7 and − 3 are combined to give − 10 .

Step 5: Standard Form Step 5: − 9 x 3 + 5 x 2 + 6 x − 10 . Finally, the polynomial is written in standard form, with the terms arranged in descending order of their exponents.


Examples
Polynomial subtraction is used in various real-world applications, such as in engineering to calculate the difference between two signal waveforms, in economics to determine the change in revenue or cost functions, and in computer graphics to manipulate and combine geometric shapes. For example, if you have two different investment portfolios and you want to know the difference in their values over time, you would subtract one polynomial function (representing one portfolio) from another (representing the other portfolio).

Answered by GinnyAnswer | 2025-07-04