Synthetic and Long division

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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 uses the coefficients of the polynomial and is generally faster than long division.

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What is long division in polynomials?

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Long division in polynomials is a method used to divide a polynomial by another polynomial, similar to numerical long division. It involves dividing, multiplying, and subtracting until the remainder is of lower degree than the divisor.

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How do you determine the order of polynomial coefficients?

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The order of polynomial coefficients is determined by the degree of the polynomial, starting from the highest degree term to the constant term. For example, in 6x^4 - 4x^3 + 0x^2 + 3x + 1, the order is 6, -4, 0, 3, 1.

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What is a zero placeholder in synthetic division?

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A zero placeholder is used in synthetic division to represent missing terms in a polynomial. For example, in the polynomial x^4 + 3x^2 - 4x - 2, the x^3 term is missing, so a zero is used as a placeholder.

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What is the remainder theorem?

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The remainder theorem states that when a polynomial f(x) is divided by (x - c), the remainder of that division is f(c). This can be used to evaluate polynomials quickly.

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What is the factor theorem?

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The factor theorem states that (x - c) is a factor of the polynomial f(x) if and only if f(c) = 0. This is useful for finding roots of polynomials.

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How do you perform synthetic division?

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To perform synthetic division, write the coefficients of the polynomial, use the root of the divisor (c), and follow the synthetic division process: bring down the leading coefficient, multiply, and add down the columns.

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What is the significance of the leading coefficient?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the end behavior of the polynomial function.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial. For example, in 2x^3 - x^2 + 5, the degree is 3.

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What does it mean for a polynomial to be in standard form?

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A polynomial is in standard form when its terms are arranged in descending order of degree, from the highest degree to the lowest.