The polynomial should be written in standard form and all of its coefficients, even those with a coefficient of 0, should be present.
- x^4+0x^3+0x^2+0x+1
Practice makes perfect
We are asked to rewrite the polynomial 1-x^4 as if we were about to divide it by x-1. Since 1-x^4 is the expression being divided, we will refer to it as the dividend. To rewrite this expression, we will complete two steps.
Rewrite the polynomial obtained in Step 1 so that all of the coefficients are present.
Let's complete these steps one at a time!
Step 1
We will start by writing 1-x^4 in standard form. In other words, we will rearrange the terms of the polynomial, so that they are arranged in decreasing order of degree.
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Polynomial: & 1-x^4
Standard Form: & - x^4+1
The dividend written in standard form helps us divide it correctly. This makes the division easier if we are using polynomial long division and it is mandatory if we are using synthetic division.
Step 2
Now, let's rewrite the polynomial obtained in Step 1 so that all of the coefficients are present. This means we will add any missing terms to the polynomial with a coefficient of 0.
- x^4+1
⇕
- x^4+ 0x^3+ 0x^2+ 0x+1
When adding the missing terms, make sure to keep the polynomial in standard form.
Extra
Performing the Division
In case you were interested, here is how we can use long division to divide 1-x^4 by x-1.
