Polynomial Functions Review

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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 does it mean to perform polynomial addition?

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Polynomial addition involves combining like terms from two or more polynomial expressions. For example, (3x^2 + 2x) + (4x^2 + 5) = (3x^2 + 4x^2) + 2x + 5 = 7x^2 + 2x + 5.

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How do you subtract polynomials?

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To subtract polynomials, distribute the negative sign to the second polynomial and then combine like terms. For example, (5x^2 + 3) - (2x^2 + 1) = (5x^2 - 2x^2) + (3 - 1) = 3x^2 + 2.

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What are the roots of a polynomial?

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The roots of a polynomial are the values of x that make the polynomial equal to zero. They can be found by factoring the polynomial or using the quadratic formula for quadratic 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. For example, in the polynomial 4x^3 + 2x^2 - x + 5, the degree is 3.

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What is the difference between even and odd functions?

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An even function satisfies f(-x) = f(x) for all x, while an odd function satisfies f(-x) = -f(x). Even functions are symmetric about the y-axis, and odd functions are symmetric about the origin.

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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 are polynomials of lower degrees. For example, x^2 - 5x + 6 can be factored as (x - 2)(x - 3).

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

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How do transformations affect the graph of a polynomial function?

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Transformations such as shifts, reflections, and stretches/compressions change the position and shape of the graph. For example, y = f(x - 2) shifts the graph 2 units to the right.

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What is the process of polynomial long division?

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Polynomial long division is a method used to divide a polynomial by another polynomial. It involves dividing the leading terms, multiplying, and subtracting until the remainder is of lower degree than the divisor.