Graphing Polynomials

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What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

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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. For example, in f(x) = 5x^4 + 2x^3 + 2x - 7, the degree is 4.

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What does it mean for a polynomial to be increasing on an interval?

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A polynomial is increasing on an interval if, for any two points x1 and x2 in that interval where x1 < x2, the value of the polynomial at x1 is less than the value at x2 (f(x1) < f(x2)).

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What is a zero of a polynomial?

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A zero of a polynomial is a value of x for which the polynomial evaluates to zero (P(x) = 0). Zeros are also known as roots or solutions.

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How can you find the y-intercept of a polynomial?

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The y-intercept of a polynomial is found by evaluating the polynomial at x = 0. For example, for f(x) = x^3 - 5x^2 - 25x + 125, the y-intercept is f(0) = 125.

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What is the maximum number of real solutions a polynomial can have?

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The maximum number of real solutions a polynomial can have is equal to its degree. For example, a polynomial of degree 4 can have at most 4 real solutions.

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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. It affects the end behavior of the polynomial's graph.

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What does it mean for a polynomial to have complex solutions?

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Complex solutions occur when the polynomial has no real solutions. These solutions are in the form of a + bi, where a and b are real numbers and i is the imaginary unit.

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How do you determine the number of zeros of a polynomial function?

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The number of zeros can be determined by the degree of the polynomial and by using the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots (counting multiplicities).

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What is the relationship between the degree of a polynomial and its graph?

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The degree of a polynomial affects the shape and number of turns in its graph. Higher degree polynomials can have more complex shapes.