The Remainder Factor Theorem and Synthetic Division

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What is the Remainder Factor Theorem?

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The Remainder Factor Theorem states that if a polynomial f(x) is divided by (x - c), the remainder of this division is f(c). This theorem helps in determining whether (x - c) is a factor of f(x).

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It is faster and requires less writing than long division.

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When using synthetic division, what do you do with the coefficients of the polynomial?

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You write down the coefficients of the polynomial in descending order of degree. If any degree is missing, use a coefficient of 0 for that degree.

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What does the remainder equal when a polynomial is divided by a factor?

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The remainder equals 0 when the polynomial is divided by a factor.

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How do you determine if (x - c) is a factor of f(x) using synthetic division?

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Perform synthetic division with c. If the remainder is 0, then (x - c) is a factor of f(x).

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What is the first step in synthetic division?

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The first step is to write down the coefficients of the polynomial in descending order.

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What is the significance of the remainder in synthetic division?

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The remainder indicates the value of the polynomial at the divisor's root. If the remainder is 0, it confirms that the divisor is a factor.

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How do you set up synthetic division for the polynomial f(x) = x^5 + 2x^4 + 9x^3 + 18x^2 + 40x + 77 divided by (x - 2)?

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Use the coefficients: 1, 2, 9, 18, 40, 77. The divisor is 2.

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What is the quotient when dividing f(x) = x^5 + 2x^4 + 9x^3 + 18x^2 + 40x + 77 by (x - 2)?

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The quotient is x^4 + 4x^3 + 17x^2 + 40x + 77.

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What is the formula for synthetic division?

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The formula for synthetic division is: 1. Write down the coefficients of the polynomial. 2. Bring down the leading coefficient. 3. Multiply and add down the column.