Polynomial Expressions and Functions Flashcard

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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 evaluates to zero. They are also known as the roots of the polynomial.

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How do you find the zeros of a cubic polynomial?

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To find the zeros of a cubic polynomial, you can use methods such as factoring, synthetic division, or the Rational Root Theorem. You can also apply numerical methods or graphing to approximate the roots.

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree $$n$$ has exactly $$n$$ complex roots, counting multiplicities.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form $$x - c$$. It is faster and requires less writing.

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

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The degree of a polynomial indicates the maximum number of zeros (roots) it can have. For example, a polynomial of degree 3 can have up to 3 zeros.

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How do you factor a polynomial completely?

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To factor a polynomial completely, look for common factors, use grouping, apply the difference of squares, or use the quadratic formula for quadratics. For higher degrees, synthetic division may help.

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What is a complex root?

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A complex root is a solution to a polynomial equation that includes imaginary numbers. Complex roots occur in conjugate pairs when the coefficients of the polynomial are real.

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What is the significance of the leading coefficient in a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial function and determines whether the graph rises or falls as $$x$$ approaches infinity.

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What is the process to write a polynomial function given its zeros?

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To write a polynomial function given its zeros, use the factored form: $$f(x) = a(x - r_1)(x - r_2)...(x - r_n)$$ where $$r_1, r_2, ..., r_n$$ are the zeros and $$a$$ is a non-zero constant.