Polynomial Function (non-calc)

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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 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 f(x) = 3x^4 + 2x^2 + 1, the degree is 4.

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What is a leading coefficient?

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

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What is a quartic function?

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A quartic function is a polynomial function of degree four, which can be expressed in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e, where a ≠ 0.

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How do you perform polynomial long division?

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To perform polynomial long division, divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by this result, subtract from the dividend, and repeat with the new polynomial until the degree of the remainder is less than the degree of the divisor.

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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 is 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 are the roots of a polynomial?

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The roots of a polynomial are the values of x for which the polynomial equals zero. They can be found by solving the equation f(x) = 0.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial division that is used when dividing by a linear factor of the form (x - c). It is quicker and involves less writing than long division.

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What is the difference between even and odd degree polynomials?

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Even degree polynomials have a U-shaped graph and can have a minimum or maximum value, while odd degree polynomials have an S-shaped graph and always cross the x-axis.