Synthetic Division

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What is synthetic division?

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Synthetic division is a simplified form of polynomial division that is used specifically for dividing a polynomial by a linear factor of the form (x - c). It is a more efficient method than long division.

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What is a potential zero of a polynomial?

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A potential zero of a polynomial is a value of x that, when substituted into the polynomial, may yield a result of zero. It is often found using the Rational Root Theorem.

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How do you determine if x = 3 is a potential zero for the polynomial x^3 - 4x^2 - 30x - 18?

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To determine if x = 3 is a potential zero, substitute 3 into the polynomial. If the result is zero, then it is a zero; otherwise, it is not.

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What are the zeros of the polynomial (x^3 + 3x^2 - 4) when divided by (x - 1)?

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The zeros are x = 1 and x = -2.

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What is the result of using synthetic division on (x^3 - 5x + 6) divided by (x - 5)?

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The result is x^2 + 5x + 20 + (106/(x - 5)).

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How can you verify if x = -3 is a solution for the polynomial f(x) = x^3 - 7x^2 - 21x + 25?

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Substitute x = -3 into the polynomial. If f(-3) equals zero, then it is a solution; otherwise, it is not.

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What is the reduced polynomial when solving (x^3 + 2x^2 - 11x - 12) divided by (x - 3)?

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The reduced polynomial is x^2 + 5x + 4.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or zero) of a polynomial equation with integer coefficients is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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What is the significance of finding zeros of a polynomial?

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Finding the zeros of a polynomial helps in understanding the behavior of the polynomial function, including its intercepts and the intervals where it is positive or negative.

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What is the difference between synthetic division and long division of polynomials?

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Synthetic division is a shortcut method specifically for dividing by linear factors, while long division can be used for any polynomial division, including non-linear divisors.