Graphing Polynomials Higher Degree

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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 P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer.

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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 P(x) = 3x^4 - 7x^2 + 2, the degree is 4.

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How do you determine where a polynomial function is positive?

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To determine where a polynomial function is positive, find the intervals where the function's value is greater than zero by analyzing the sign of the function in each interval defined by its roots.

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What is the end behavior of a polynomial function?

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The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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What does it mean for a polynomial to be factored?

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Factoring a polynomial means expressing it as a product of its factors, which can be linear or quadratic expressions. For example, x^2 - 5x + 6 can be factored as (x-2)(x-3).

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What is a root of a polynomial?

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A root of a polynomial is a value of x for which the polynomial equals zero. For example, if P(x) = x^2 - 4, the roots are x = 2 and x = -2.

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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 function.

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How can you identify the number of turning points in a polynomial graph?

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The maximum number of turning points of a polynomial function is one less than its degree. For example, a polynomial of degree 4 can have up to 3 turning points.

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What is the difference between even and odd degree polynomials?

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Even degree polynomials have symmetric end behavior (both ends go up or both go down), while odd degree polynomials have opposite end behavior (one end goes up and the other goes down).

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its linear factors. For example, y = (x-1)(x+2) is a factored form of a quadratic polynomial.