Use synthetic division. `(x^5-1)/(x-1)`

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lemjay eNotes educator| Certified Educator


Before doing the synthetic division, consider the degree of the polynomial at the numerator. Since its degree is 5, it should have 6 terms.
`=(x^5+0x^4+0x^3+0x^2+0x-1) /(x-1)`

Now that all the terms of the polynomial are included, proceed to divide. In synthetic division, the coefficients and constant of the polynomial at the numerator are the dividends. While the root of the denominator is the divisor.

The root of x-1 is:




`1`    `|`    `1`    `0`    `0`    `0`    `0`    `-1`
                  `1`     `1`      `1`     `1`        `1`  
           `1`     `1`     `1`      `1`      `1`       `0`

The numbers at the last row are the coefficients of the resulting polynomial  which is one degree less than the numerator.

Hence, `(x^5-1)/(x-1)=x^4+x^3+x^2+x+1` .