Synthetic Division

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What is Synthetic Division?

Back

Synthetic Division is a simplified method of dividing a polynomial by a linear binomial of the form (x - c). It is faster than long division and is particularly useful for finding polynomial roots.

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What is the first step in Synthetic Division?

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The first step is to set up the synthetic division by writing down the coefficients of the polynomial and the value of c from the divisor (x - c).

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How do you handle coefficients of zero in Synthetic Division?

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If there are missing degrees in the polynomial, represent their coefficients as zero. For example, for the polynomial 2x^3 + 0x^2 + 3x + 4, the coefficients are [2, 0, 3, 4].

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What is the purpose of the remainder in Synthetic Division?

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The remainder indicates the value of the polynomial at the root c. If the remainder is zero, it means that (x - c) is a factor of the polynomial.

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How do you find the quotient in Synthetic Division?

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The quotient is found by writing down the results of the synthetic division process, which represent the coefficients of the resulting polynomial.

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What is the relationship between Synthetic Division and the Remainder Theorem?

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The Remainder Theorem states that the remainder of the division of a polynomial f(x) by (x - c) is equal to f(c). Synthetic Division provides a method to find this remainder efficiently.

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What is the significance of the leading coefficient in Synthetic Division?

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The leading coefficient of the polynomial affects the degree of the resulting polynomial after division. It is the first number in the synthetic division process.

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How do you perform Synthetic Division for the polynomial 2x^3 + 3x^2 + 8x + 7 by (x - 4)?

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1. Write down the coefficients: [2, 3, 8, 7]. 2. Use 4 for synthetic division. 3. Bring down the leading coefficient (2). 4. Multiply and add down the column to find the quotient and remainder.

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What is the result of dividing 2x^3 + 3x^2 + 8x + 7 by (x - 4)?

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The result is a quotient of 2x^2 + 11x + 52 and a remainder of 219.

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What is the formula for the Remainder Theorem?

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If f(x) is a polynomial and c is a constant, then the remainder of f(x) divided by (x - c) is f(c).