Pre-Calculus HWK Section 2.4 Zeros of 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 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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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 P(x) = 0. They are also known as roots or solutions 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, you can use methods such as factoring, the Rational Root Theorem, synthetic division, or numerical methods like the Newton-Raphson method.

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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 indicates the maximum number of zeros the polynomial can have.

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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 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, and it must have at least one complex zero if n is greater than 0.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or zero) of a polynomial equation with integer coefficients is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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What does it mean for a polynomial to have rational coefficients?

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A polynomial has rational coefficients if all the coefficients (the numbers in front of the variables) are rational numbers, which include integers, fractions, and terminating or repeating decimals.

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How can you determine the number of positive and negative zeros of a polynomial?

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You can use Descartes' Rule of Signs: the number of positive zeros is equal to the number of sign changes in P(x), and the number of negative zeros is equal to the number of sign changes in P(-x).

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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.