Math 3 EOC Review: Synthetic Division and Polynomials

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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 used primarily for polynomials.

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What does the term 'multiplicity' refer to in polynomials?

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Multiplicity refers to the number of times a particular root (or zero) appears in a polynomial. For example, in the polynomial (x - 2)², the root x = 2 has a multiplicity of 2.

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How do you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function is determined by its leading term. If the leading coefficient is positive and the degree is even, the ends of the graph will rise. If the leading coefficient is negative and the degree is even, the ends will fall. If the degree is odd, the ends will go in opposite directions.

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What is the significance of the roots of a polynomial?

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The roots of a polynomial are the values of x that make the polynomial equal to zero. They are important for determining the x-intercepts of the graph and the behavior of the function.

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How do you find the intervals where a polynomial function is positive or negative?

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To find the intervals where a polynomial is positive or negative, identify the roots, test intervals between the roots, and determine the sign of the polynomial in those intervals.

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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 helps in finding factors 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 (x - c), and follow the synthetic division process to find the quotient and remainder.

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What is the standard form of a polynomial?

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The standard form of a polynomial is expressed as a sum of terms in descending order of the degree, such as a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.

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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. It determines the polynomial's end behavior and the maximum number of roots.