HA2 Unit 6 Test Review

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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 P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer.

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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 the variable. This can involve factoring, using the quadratic formula, or synthetic division.

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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 quicker and requires less writing.

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What is the Remainder Theorem?

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The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), the remainder of that division is f(c). This helps in finding zeros of the polynomial.

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What is the Factor Theorem?

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The Factor Theorem states that (x - c) is a factor of the polynomial f(x) if and only if f(c) = 0. This is used to find factors and zeros of polynomials.

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How do you evaluate a polynomial function at a given point?

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To evaluate a polynomial function f(x) at a point x = a, substitute a into the polynomial and simplify to find f(a).

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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. It determines the polynomial's behavior and the number of possible zeros.

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What are complex zeros?

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Complex zeros are solutions to polynomial equations that include imaginary numbers. They occur in conjugate pairs when the coefficients of the polynomial are real.

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What is the significance of the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root. This implies that a polynomial of degree n has exactly n roots, counting multiplicities.

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How do you find a polynomial given its zeros?

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To find a polynomial given its zeros, use the fact that if r1, r2, ..., rn are the zeros, the polynomial can be expressed as P(x) = k(x - r1)(x - r2)...(x - rn), where k is a constant.