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Polynomial

Factor and Remainder Theorem

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Factor The expression x-a is a linear factor of a polynomial if and only if the value a is a zero related polynomial function. Remainder If a polynomial function P(x) of a degree greater than or equal to 1 is divided by the linear factor (x-a), where a is a constant, then the remainder is P(a)

Factor and Remainder Theorem

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Factor The expression x-a is a linear factor of a polynomial if and only if the value a is a zero related polynomial function. Remainder If a polynomial function P(x) of a degree greater than or equal to 1 is divided by the linear factor (x-a), where a is a constant, then the remainder is P(a)

Steps for Factoring Polynomials

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1. Pull out greatest common factor or GCF [This is a very important step that is often skipped] Ex: or 2. Look at number of terms a. 2 Terms: If there are two terms (binomial), then check to see if it is i. A difference of squares: Ex: ii. A difference of cubes Ex: iii. A sum of cubes Ex: If there are two terms and the expression is none of the above, it is prime! b. 3 Terms: If there are three terms (trinomial), then factor into two binomials thinking of the FOIL method in reverse. Find factors of c that add up to b. Start by identifying the factors of c. The sign before c determines whether signs are the same or different in the 2 binomials that are produced, .

Definition and Domain of Rational Functions

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Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. f(x) = P(x) / Q(x) Here are some examples of rational functions: g(x) = (x2 + 1) / (x - 1) h(x) = (2x + 1) / (x + 3) The rational functions to explored in this tutorial are of the form f(x) = (ax+b)/(cx + d) where a, b, c and d are parameters that may be changed, using sliders, to understand their effects on the properties of the graphs of rational functions defined above. Example: Find the domain of each function given below. g(x) = (x - 1) / (x - 2) h(x) = (x + 2) / x Solution

Factoring Polynomials of Degree 3

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Factoring Polynomials of Degree 3 Factoring a 3 - b 3 An expression of the form a 3 - b 3 is called a difference of cubes. The factored form of a 3 - b 3 is (a - b)(a 2 + ab + b 2) : (a - b)(a 2 + ab + b 2) = a 3 - a 2 b + a 2 b - ab 2 + ab 2 - b 3 = a 3 - b 3 For example, the factored form of 27x 3 - 8 ( a = 3x, b = 2 ) is (3x - 2)(9x 2 + 6x + 4) . Similarly, the factored form of 125x 3 -27y 3 ( a = 5x, b = 3y ) is (5x - 3y)(25x 2 +15xy + 9y 2) . To factor a difference of cubes, find a and b and plug them into (a - b)(a 2 + ab + b 2) . Factoring a 3 + b 3 An expression of the form a 3 + b 3 is called a sum of cubes. The factored form of a 3 + b 3 is (a + b)(a 2 - ab + b 2) : (a + b)(a 2 - ab + b 2) = a 3 + a 2 b - a 2 b - ab 2 + ab 2 + b 3 = a 3 - b 3 .

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