To demonstrate polynomial subtraction correctly, we first need to make sure both sides of the equation are simplified and combined correctly. Let's analyze the example given:
You wrote: \(0(2x^2 + 62 + 1) - (1322 + 32 + 9) = 22 + 3x + 8\)
However, there are a few issues with this expression, such as the use of "0" and improper parentheses. Let's clarify how to perform polynomial subtraction step-by-step.
Let's define two polynomials:
- Let \(P(x) = 2x^2 + 62 + 1\)
- Let \(Q(x) = 1322 + 32 + 9\)
To subtract \(Q(x)\) from \(P(x)\): \[ P(x) - Q(x) = (2x^2 + 62 + 1) - (1322 + 32 + 9) \]
First, simplify \(Q(x)\): \[ Q(x) = 1322 + 32 + 9 = 1363 \]
Now substitute back into the equation: \[ P(x) - Q(x) = (2x^2 + 62 + 1) - 1363 \] Resulting in: \[ = 2x^2 + 63 - 1363 = 2x^2 - 1300 \]
Now, the equation becomes: \[ P(x) - Q(x) = 2x^2 - 1300 \]
This does not equal \(22 + 3x + 8\) (or \(30 + 3x\)).
To correct it, a more accurate subtraction should look like this:
For instance, if we define:
- \(P(x) = 2x^2 + 3x + 10\)
- \(Q(x) = x^2 + 6x + 2\)
Then we can do: \[ P(x) - Q(x) = (2x^2 + 3x + 10) - (x^2 + 6x + 2) \] Simplifying: \[ = (2x^2 - x^2) + (3x - 6x) + (10 - 2) \] \[ = x^2 - 3x + 8 \]
So a correct example of polynomial subtraction is indeed: \[ (2x^2 + 3x + 10) - (x^2 + 6x + 2) = x^2 - 3x + 8 \]
Let me know if you need further assistance or clarification!
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