Finding Zeros of Polynomial Functions

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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 f(x) equals zero. It represents the x-intercepts of the graph of the function.

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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 quadratic formula, synthetic division, or graphing.

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What is the y-intercept of a polynomial function?

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The y-intercept of a polynomial function is the value of the function when x = 0. It is the point where the graph intersects the y-axis.

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What does the Fundamental Theorem of Algebra state?

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The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root, and the total number of roots (including multiplicities) is equal to the degree of the polynomial.

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If a polynomial has a degree of 4, how many zeros can it have?

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A polynomial of degree 4 can have up to 4 zeros, including real and complex roots.

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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).

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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 formulas such as the difference of squares or the quadratic formula.

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What is the quadratic formula?

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The quadratic formula is used to find the roots of a quadratic equation ax^2 + bx + c = 0 and is given by x = (-b ± √(b² - 4ac)) / (2a).

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What is the significance of complex roots in polynomials?

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Complex roots occur in conjugate pairs and indicate that the polynomial does not intersect the x-axis at those points. They are important for understanding the behavior of the polynomial.

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

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You can determine the number of real roots by analyzing the discriminant (b² - 4ac) for quadratics or using Descartes' Rule of Signs for higher-degree polynomials.