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Download Intermediate Algebra Section 5.3 – Dividing Polynomials
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Transcript
Intermediate Algebra Section 5.3 – Dividing Polynomials When dividing a polynomial by a monomial, we can use the property of addition of fractions with common denominators. Dividing a Polynomial by a Monomial Divide each term in the polynomial by the monomial. a+b a b = + , where c ≠ 0 . c c c Example: Divide the following. a) −28 x 4 y 12 x 2 y 3 b) 6x − 3x by 3x 4 3 2 Section 5.3 – Dividing Polynomials c) page 2 4x 3 y + 12x 2 y 2 − 4xy 3 4 xy To divide a polynomial by a polynomial other than a monomial, we use a method called long division. This method is similar to long division of real numbers. When using long division for polynomials, the polynomials must be written in descending order. Example: a) Divide the following. 34 9 7 b) Divide 5x − 5x + 2x + 20 by x + 4 . 2 3 Section 5.3 – Dividing Polynomials c) (18x 3 + x 2 − 90x − 5)÷ (9x 2 − 45) Definition If f and g are two functions, f is the function defined by The quotient g f ( x) f , g ( x) ≠ 0 . ( x) = g ( x) g Given the functions f ( x ) = x 2 − x − 12 and g ( x ) = x − 4 , find f f ( x ) and ( 2 ) g g page 3 Section 5.3 – Dividing Polynomials page 4 The Remainder Theorem Let f be a polynomial function. If f ( x ) is divided by x − c , then the remainder is f ( c ) . Example: Use the Remainder Theorem to find the remainder when f ( x ) = x 2 + 4 x − 5 is divided by x + 2 . Section 5.3 – Dividing Polynomials page 5 The factor Theorem Let f be a polynomial function. Then x − c is a factor of f ( x ) if and only if f ( c ) = 0 . Example: Use the Factor Theorem to determine whether x + 2 is a factor of f ( x ) = 3 x 2 + x − 2 . If x + 2 is a factor, then write f ( x ) in factored form.