Alg2 Review on Section 3.4

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Define 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 $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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What is a zero of a polynomial function?

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A zero of a polynomial function is a value of $$x$$ for which the function evaluates to zero, i.e., $$f(x) = 0$$. Zeros are also known as roots 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, synthetic division, or the Rational Root Theorem, and then solve for $$x$$.

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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 than long division.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation with integer coefficients is of the form $$\frac{p}{q}$$, where $$p$$ is a factor of the constant term and $$q$$ is a factor of the leading coefficient.

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Factor the polynomial $$x^2 - 5x + 6$$.

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The polynomial can be factored as $$(x - 2)(x - 3)$$.

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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 at most $$n$$ real zeros, counting multiplicities.

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What is long division of polynomials?

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Long division of polynomials is a method used to divide a polynomial by another polynomial of lower degree, similar to numerical long division, resulting in a quotient and a remainder.

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Solve for zeros: $$f(x) = x^3 - 6x^2 + 11x - 6$$.

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The zeros are $$x = 1, 2, 3$$.

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What is a factor of a polynomial?

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A factor of a polynomial is a polynomial of lower degree that divides the original polynomial without leaving a remainder.