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

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What are the zeros of a polynomial function?

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The zeros of a polynomial function are the values of \(x\) for which the function equals zero. They are also known as the roots of the polynomial.

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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 factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its linear factors. For example, if the zeros of a polynomial are \(x = r_1, r_2, ..., r_n\), the factored form is: $$f(x) = a(x - r_1)(x - r_2)...(x - r_n)$$ where \(a\) is a leading coefficient.

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

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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 \(f(x) = 3x^4 + 2x^3 - x + 5\), the leading coefficient is 3.

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What is multiplicity in relation to polynomial zeros?

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Multiplicity refers to the number of times a particular zero appears in a polynomial. For example, if \(x = 2\) is a zero of the polynomial \(f(x) = (x - 2)^3(x + 1)\), then it has a multiplicity of 3.

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What is a quartic function?

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A quartic function is a polynomial function of degree 4. It can be expressed in the form: $$f(x) = ax^4 + bx^3 + cx^2 + dx + e$$ where \(a \neq 0\).

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

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A polynomial of degree \(n\) can have up to \(n\) real zeros. Some of these zeros may be repeated (multiplicity) or complex.

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How do you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function is determined by its leading term. If the leading coefficient is positive, the function rises to the right; if negative, it falls to the right. The degree also affects whether it rises or falls to the left.