# Use synthetic division to solve (x^{4} - 1) ÷ (x - 1). What is the quotient?

**Solution:**

Given: (x^{4} - 1) ÷ (x - 1)

When the divisor is a linear factor we can use synthetic division of polynomials.

- We write the coefficients alone as the dividend.
- We take the zero of the divisor.
- Do multiplication and addition instead of the division and subtraction done in the long division method.

The quotient obtained by this method is x^{3} +x^{2} +x +1 and the remainder is 0.

Let us verify the calculation:

We know that, (a + b)(a - b) = (a^{2} - b^{2})

So,

(x^{4} - 1) = (x^{2} - 1) (x^{2} + 1) = (x + 1)(x - 1)(x^{2} + 1)

By simplification we get,

(x^{4} - 1) ÷ (x - 1) = [(x + 1)(x - 1)(x^{2} + 1)]/(x - 1)

= (x + 1)(x^{2} + 1)

= x^{3} + x + x^{2} + 1

Therefore, the quotient is (x^{3} + x^{2} + x + 1).

## Use synthetic division to solve (x^{4} - 1) ÷ (x - 1). What is the quotient?

**Summary:**

Use synthetic division to solve (x^{4} - 1) ÷ (x - 1). The quotient is (x^{3} + x + x^{2} + 1).

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