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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n \neq 0\) 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 $$y = -4x^5 + 3x^2 - 8x + 1$$, the degree is 5.

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How do you find the roots of a polynomial?

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The roots of a polynomial are the values of \(x\) that make the polynomial equal to zero. They can be found by factoring the polynomial or using the quadratic formula for polynomials of degree 2.

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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, the polynomial with roots \(x = -2, x = 5, x = 3\) can be factored as $$y = (x + 2)(x - 5)(x - 3)$$.

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What does multiplicity mean in the context of polynomial roots?

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Multiplicity refers to the number of times a particular root appears in a polynomial. For example, in the polynomial $$y = (x + 2)(x - 7)^3$$, the root \(-2\) has a multiplicity of 1, and \(7\) has a multiplicity of 3.

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How do you determine the degree of a polynomial from its graph?

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The degree of a polynomial can be determined from its graph by observing the number of turning points. A polynomial of degree \(n\) can have at most \(n-1\) turning points.

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What is the leading coefficient of a polynomial?

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

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What is the standard form of a polynomial?

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The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, $$y = 2x^3 + 3x^2 - 5x + 1$$ is in standard form.

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What is the difference between a monomial, binomial, and trinomial?

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A monomial is a polynomial with one term (e.g., $$3x^2$$), a binomial has two terms (e.g., $$x^2 + 4$$), and a trinomial has three terms (e.g., $$x^2 + 4x + 4$$).

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How do you add and subtract polynomials?

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To add or subtract polynomials, combine like terms. For example, $$ (2x^2 + 3x) + (4x^2 - 2x) = 6x^2 + x $$.