Use synthetic division to solve (x⁴ - 1) ÷ (x - 1). What is the quotient?

Solution:

Given: (x⁴ - 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 x³ +x² +x +1 and the remainder is 0.

Let us verify the calculation:

We know that, (a + b)(a - b) = (a² - b²)

So,

(x⁴ - 1) = (x² - 1) (x² + 1) = (x + 1)(x - 1)(x² + 1)

By simplification we get,

(x⁴ - 1) ÷ (x - 1) = [(x + 1)(x - 1)(x² + 1)]/(x - 1)

= (x + 1)(x² + 1)

= x³ + x + x² + 1

Therefore, the quotient is (x³ + x² + x + 1).

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Use synthetic division to solve (x⁴ - 1) ÷ (x - 1). What is the quotient?

Summary:

Use synthetic division to solve (x⁴ - 1) ÷ (x - 1). The quotient is (x³ + x + x² + 1).