5.1 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 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. For example, in the polynomial 4x^3 + 2x^2 - x + 7, the degree is 3.

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How do you factor a polynomial?

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To factor a polynomial, look for the greatest common factor (GCF) first, then apply methods such as grouping, using special products (like difference of squares), or trial and error for quadratic polynomials.

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What is the GCF in the polynomial 6x^3 + 9x^2?

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The GCF is 3x^2.

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What is a turning point in a polynomial graph?

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A turning point is a point where the graph changes direction from increasing to decreasing or vice versa. The maximum number of turning points of a polynomial function is one less than its degree.

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What is the maximum number of turning points for a polynomial of degree 4?

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The maximum number of turning points is 3.

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What is the difference between a linear and a quadratic polynomial?

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A linear polynomial is of degree 1 (e.g., f(x) = mx + b), while a quadratic polynomial is of degree 2 (e.g., f(x) = ax^2 + bx + c).

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What does it mean for a polynomial to be factored completely?

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A polynomial is factored completely when it is expressed as a product of irreducible polynomials, meaning it cannot be factored further over the set of real numbers.

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What is the standard form of a polynomial?

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The standard form of a polynomial is written in descending order of the degree of its terms, such as f(x) = ax^n + bx^(n-1) + ... + k.

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What is the leading coefficient of the polynomial 5x^4 - 3x^3 + 2?

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The leading coefficient is 5, which is the coefficient of the term with the highest degree.