4.5 Polynomial Equations Check-Up

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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, in the form of a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient.

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How do you use synthetic division to test for possible zeros of a polynomial?

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To use synthetic division, set up the synthetic division with the possible zero on the left and the coefficients of the polynomial on the right. Perform the division and check the remainder. If the remainder is zero, then the number is a zero of the polynomial.

Front

What is a polynomial function?

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Front

What does it mean for a polynomial to have a zero at x = c?

Back

Front

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 determines the end behavior of the polynomial function as x approaches positive or negative infinity.

Front

What is the difference between a polynomial and a rational function?

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A polynomial is a function that consists of terms with non-negative integer exponents, while a rational function is a ratio of two 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. It indicates the maximum number of roots the polynomial can have.

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How can you determine the number of possible real roots of a polynomial?

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The number of possible real roots of a polynomial can be determined by its degree. A polynomial of degree n can have at most n real roots.

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What is a multiplicity of a root?

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The multiplicity of a root refers to the number of times a particular root appears in the factorization of a polynomial. A root with an even multiplicity will touch the x-axis, while a root with an odd multiplicity will cross the x-axis.

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What is the relationship between the factors and zeros of a polynomial?

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The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero, and these values are also the roots of the polynomial's factors.