Synthetic Division with Missing Coefficient

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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 binomial of the form (x - c). It is faster than long division and is particularly useful when the divisor is a first-degree polynomial.

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What is the purpose of finding the value of h in synthetic division problems?

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The value of h is determined to ensure that the polynomial division results in a specific quotient and remainder, allowing for the equation to hold true for values of x not equal to the root of the divisor.

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How do you set up synthetic division for a polynomial with a missing coefficient?

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List the coefficients of the polynomial in descending order of degree, inserting zeros for any missing coefficients. Then, use the synthetic division process with the root of the divisor.

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What is the remainder theorem in relation to synthetic division?

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The remainder theorem states that if a polynomial f(x) is divided by (x - c), the remainder of this division is f(c). This is useful for finding specific values of polynomials.

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Set the coefficients of the corresponding terms equal to each other. For this equation, h must equal 4 to satisfy the equality.

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What is the significance of the divisor in synthetic division?

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The divisor determines the value of x for which the polynomial is evaluated. It also influences the coefficients of the resulting polynomial after division.

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By equating the coefficients of the polynomial on both sides, you find that h must equal 2.

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What is the process for synthetic division?

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1. Write down the coefficients of the polynomial. 2. Use the root of the divisor. 3. Bring down the leading coefficient. 4. Multiply and add down the column. 5. The last number is the remainder.

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Set the coefficients equal, leading to h = 3.

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What is the role of the remainder in synthetic division?

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The remainder indicates what is left over after division. If the remainder is zero, it means the polynomial is divisible by the divisor without any leftover.