Advanced Polynomial Mathematics Examples and Theorems
Explore long division, synthetic division, remainder theorem, rational zeros theorem, and finding rational zeros of polynomials with detailed examples and explanations. Enhance your understanding of polynomial functions and algebraic concepts.
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Long division and the division algorithm. Example 1: evaluate the following: I. A. 3 2 2x + 1 x 2 5 x ) + 3 2 2 5 4 ( 8 7 2 x x x x x 3 2 4 10 x + ) + + 2 2 2 7 5 2 ( 2 2 x x x x x x ( 2 2 5) 3 + = + 1) 3 3 2 2 4 8 7 2 (2 5)(2 x x x x x x
B. Example 2: Use long division to find the quotient and remainder when is divided by 2 3 2 x x x + + + 2 2 1 x x 4 3 1 + 2 x 2 x + 2 1 x x + 2 2 1 x from the division algorithm: + ( )( ) ( 1 + ) + = + + 4 3 2 2 2 2 3 2 1 2 1 2 2 x x x x x x x x In fraction form: 2 3 2 x + + + + 4 3 2 1 2 x 2 x x x x = + 2 1 x x + 2 2 1 2 1 x x
II. Synthetic division. A. Example 4: Divide the following using synthetic division. Write a summary statement in fraction form. 2 3 6 8 28 3 4 14 18 2 6 10 + 3 2 3 2 6 10 x x x x 2 + 3 2 3 2 6 10 18 x x x x x = + + + 2 3 4 14 x x 2 2
B. Remainder Theorem If a polynomial f(x) is divided by x-k, then the remainder is . ( ) r f k = C. Example 3: Find the remainder when is divided by: = + 2 ( ) f x 2 4 1 x x 1. x+ 1 ( 1) f = 2( 1) 4( 1) 1 + = 2 7 2. x 5 = + = 2 (5) 2(5) 4(5) 1 31 f 3. x + 3 ( 3) f = 2( 3) 4( 3) 1 + = 2 31
IV. Rational Zeros Theorem: A. rational or irrational. Real zeros of polynomial functions can be either B. Theorem: Suppose f is a polynomial function of degree n > 0 of the form: ( ) f x = + + + 1 n n ... a x a x a 1 0 n n with every coefficient an integer and . If x=p/q is a rational zero of f, where p and q have no common integer factors other than 1, then a 0 0 p is an integer factor of the constant coefficient a0. q is an integer factor of the leading coefficient an.
C. Example 5: List all possible zeros of the following functions: = + 3 a. ( ) g x 3 2 4 x x 4 3 2 3 1 3 4 1 2 1 1 1 4 3 2 3 1 3 , 4, 2, 1 , , , , , , , = + 4 3 b. ( ) h x 6 2 5 x x x 5 6 5 3 5 2 5 1 1 6 1 3 1 2 1 1 5 6 5 3 5 2 1 6 1 3 1 2 , 5, , 1 , , , , , , , , , , ,
D. Example 6: finding the rational zeros. Find the rational zeros of each of the following polynomials: = + + 3 2 ( ) f x 2 13 6 x x x 1 1. x = 2, 3,2 = + 3 2 2. ( ) g x 4 1 x x no rational zeros
Example 7: Write the equation in standard form for the quartic polynomial with the zeros 1, -1, 2, and -1/2, and through the point (-2,9) ( ) ( ) 1 ( f x a x x = + ( ) ( 2) 9 ( 2) 1 (( 2) 1)(( 2) 2)(2( 2) 1) a = = + 9 36a = 1 4 ( ) ( ) 1 ( 4 E. + 1)( 2)(2 1) x x + f = a 1 = + + 1)( 2)(2 1) f x x x x x
