Polynomial Functions & Equations

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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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 f(x) = 0, then x is a zero of the polynomial.

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What is the Factor Theorem?

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The Factor Theorem states that if f(c) = 0 for a polynomial f(x), then (x - c) is a factor of f(x).

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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).

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the function equal to zero and solve for x. This may involve factoring, using the quadratic formula, or other algebraic methods.

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What is the multiplicity of a zero?

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The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. A zero with an even multiplicity will touch the x-axis, while a zero with an odd multiplicity will cross the x-axis.

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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 is faster than long division and is used 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 written in descending order of the powers of x. For example, f(x) = 4x^3 - 2x^2 + 3x - 5.

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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 the polynomial 3x^4 - 2x^2 + 1, the degree is 4.

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What is the leading coefficient of a polynomial?

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The leading coefficient of a polynomial is the coefficient of the term with the highest degree. For example, in 5x^3 + 2x^2 - 3, the leading coefficient is 5.