Algebra 2 Polynomial Functions 3.1, 3.5

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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 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 f(x) = 4x^3 + 2x^2 - x + 7, the degree is 3.

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What is a leading term in a polynomial?

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The leading term of a polynomial is the term with the highest degree. For example, in f(x) = 5x^4 + 3x^2 - 2, the leading term is 5x^4.

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What is a leading coefficient?

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

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How do you determine 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. The solutions are the x-values where the graph intersects the x-axis.

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What does it mean for a zero to have multiplicity?

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Multiplicity refers to the number of times a particular zero appears as a solution to the polynomial equation. For example, if (x - 2)^3 is a factor, then x = 2 has a multiplicity of 3.

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What is the relationship between the degree of a polynomial and the number of possible zeros?

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The degree of a polynomial indicates the maximum number of possible real zeros it can have. For example, a polynomial of degree 4 can have up to 4 real zeros.

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What is the significance of the leading coefficient in determining the end behavior of a polynomial function?

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The leading coefficient determines the direction of the graph as x approaches positive or negative infinity. If the leading coefficient is positive, the graph rises to the right; if negative, it falls 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 the terms. For example, f(x) = 2x^4 - 3x^3 + x - 5 is in standard form.

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How do you classify polynomials based on their degree?

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Polynomials are classified as follows: Constant (degree 0), Linear (degree 1), Quadratic (degree 2), Cubic (degree 3), Quartic (degree 4), and so on.