Polynomial Functions

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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, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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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 expression. For example, in the polynomial 4x^3 + 2x^2 - x + 7, the degree is 3.

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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 polynomial equal to zero and solve for the variable. This can involve factoring, using the quadratic formula, or applying synthetic division.

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What is the difference between a factor and a root of a polynomial?

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A factor of a polynomial is a polynomial that divides the original polynomial without leaving a remainder, while a root (or zero) is a value of the variable that makes the polynomial equal to zero.

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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 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.

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What transformations occur when a polynomial is shifted right by 3 units?

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Shifting a polynomial function f(x) to the right by 3 units results in the new function f(x - 3). This means every point on the graph of f(x) moves 3 units to the right.

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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 degree of its terms. For example, 2x^4 + 3x^3 - x + 5 is in standard form.

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How do you add two polynomial functions?

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To add two polynomial functions, combine like terms by adding the coefficients of terms with the same degree.

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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 - 4, the leading coefficient is 5.