Polynomial Review

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What is a polynomial function?

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A polynomial function is a mathematical expression that involves 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 \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.

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What are the zeros of a polynomial?

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The zeros of a polynomial are the values of \(x\) for which the polynomial evaluates to zero. They are also known as roots or solutions of the polynomial equation.

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How do you find 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 \(f(x) = 2x^3 + 3x^2 - x + 5\), the degree is 3.

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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 factors. For example, the polynomial \(f(x) = x^2 - 5x + 6\) can be factored as \(f(x) = (x - 2)(x - 3)\).

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What is the relationship between zeros and factors of a polynomial?

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If \(r\) is a zero of a polynomial \(f(x)\), then \(f(x)\) can be factored as \(f(x) = (x - r)g(x)\), where \(g(x)\) is another polynomial.

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How do you determine the least degree polynomial given its zeros?

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To determine the least degree polynomial given its zeros, create factors for each zero. For example, if the zeros are 1, -4, and 8, the polynomial can be expressed as \(f(x) = (x - 1)(x + 4)(x - 8)\).

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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, \(f(x) = 4x^3 - 2x^2 + 3x - 5\) 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., \(x^2 + 4x\)). A trinomial has three terms (e.g., \(x^2 + 4x + 4\)).

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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 \(f(x) = 5x^4 + 3x^3 - 2x + 1\), the leading coefficient is 5.

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How do you add two polynomials?

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To add two polynomials, combine like terms. For example, \(f(x) = 2x^2 + 3x + 1\) and \(g(x) = x^2 - 4x + 2\) can be added as follows: \(f(x) + g(x) = (2x^2 + x^2) + (3x - 4x) + (1 + 2) = 3x^2 - x + 3\).