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 the polynomial f(x) = 2x^3 - 4x + 1, the degree is 3.

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How do you determine the number of solutions for a polynomial equation?

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The number of solutions (or roots) of a polynomial equation can be determined by its degree. A polynomial of degree n can have up to n real solutions.

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What is the quadratic formula?

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

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What is the maximum number of turns a polynomial graph can have?

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The maximum number of turns a polynomial graph can have is one less than its degree. For example, a polynomial of degree 4 can have at most 3 turns.

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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 that makes the polynomial equal to zero. For example, if f(x) = x^2 - 4, then the roots are x = 2 and x = -2.

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What is the leading coefficient of a polynomial?

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The leading coefficient of a polynomial is the coefficient of the term with the highest degree. For example, in f(x) = 3x^4 + 2x^3 - x + 5, the leading coefficient is 3.

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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. For example, the polynomial x^2 - 5x + 6 can be factored as (x - 2)(x - 3).

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What is the significance of the discriminant in a quadratic equation?

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The discriminant (D) in a quadratic equation ax^2 + bx + c = 0 is given by D = b² - 4ac. It determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

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How do you find the vertex of a quadratic function?

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The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula: x = -b/(2a). The y-coordinate can be found by substituting this x value back into the function.