Finding 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 f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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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. For example, in f(x) = 3x^4 - 2x^2 + 5, the degree is 4.

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How do you find the y-intercept of a polynomial function?

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To find the y-intercept of a polynomial function, evaluate the function at x = 0. The y-intercept is the point (0, f(0)).

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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 f(x) = 0. It represents the x-coordinate where the graph of the function intersects the x-axis.

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How many zeros can a polynomial function have?

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A polynomial function of degree n can have at most n real zeros, counting multiplicities.

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What is the Factor Theorem?

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The Factor Theorem states that if f(c) = 0 for a polynomial f(x), then (x - c) is a factor of f(x).

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - c). It is faster than long division and is used to find zeros.

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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 ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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How do you factor a polynomial?

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To factor a polynomial, look for common factors, use grouping, or apply special factoring techniques such as difference of squares or perfect square trinomials.

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What is the relationship between the zeros and the factors 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 each zero (x - c) represents a factor of the polynomial.