Solving polynomials

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to whole number powers and coefficients. For example, P(x) = 2x^3 - 4x^2 + 3x - 5 is a polynomial.

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What does it mean for a number to be a zero of a polynomial?

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A zero of a polynomial is a value of x that makes the polynomial equal to zero. In other words, if P(c) = 0, then c is a zero of the polynomial P.

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How do you find the zeros of a polynomial given a factor?

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If you know that (x - c) is a factor of the polynomial P(x), you can find the zeros by performing polynomial long division or synthetic division to divide P(x) by (x - c), and then solving the resulting polynomial for its zeros.

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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 equal to f(c).

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What is synthetic division?

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

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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 P(x) if and only if P(c) = 0.

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How do you list the zeros of a polynomial function?

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To list the zeros of a polynomial function, factor the polynomial completely, set each factor equal to zero, and solve for x.

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What is the significance of the degree of a polynomial?

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The degree of a polynomial indicates the highest power of the variable in the polynomial. It determines the number of zeros (real and complex) the polynomial can have.

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What is a binomial?

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A binomial is a polynomial with exactly two terms, such as (x + 1) or (3x^2 - 2).

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How do you determine if a binomial is a factor of a polynomial?

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To determine if a binomial (x - c) is a factor of a polynomial P(x), you can use the Factor Theorem: if P(c) = 0, then (x - c) is a factor.