Zeros of Polynomials

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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 the 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 factor the polynomial and set each factor equal to zero, or use methods such as synthetic division or the quadratic formula for quadratic polynomials.

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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 when it is written in standard form.

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What does it mean for a polynomial to have a multiplicity?

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Multiplicity refers to the number of times a particular zero is repeated in a polynomial. For example, if (x - 2) appears twice in the factorization, then 2 is a zero with a multiplicity of 2.

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What is the standard form of a polynomial?

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The standard form of a polynomial is when it is written in descending order of the degree of its terms, from the highest degree to the lowest.

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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.

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What is the difference between even and odd functions?

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An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. For polynomials, even degree indicates an even function and odd degree indicates an odd function.

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How can you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function can be determined by the degree and the leading coefficient. If the degree is even and the leading coefficient is positive, both ends go up; if negative, both ends go down.

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What is factor by grouping?

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Factor by grouping is a method used to factor polynomials with four or more terms by grouping terms in pairs and factoring out common factors.

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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 is a factor of the constant term divided by a factor of the leading coefficient.