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Division and Factors • When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor of the dividend. Example: Divide to determine whether x + 3 and x 1 are factors of 3 2 x 2 x 5 x 4. Division and Factors continued • Divide: x 2 5 x 20 x3 x3 2 x 2 5 x 4 x3 3x 2 5 x 2 5 x 5 x 2 15 x 20 x 4 20 x 60 64 remainder Since the remainder is –64, we know that x + 3 is not a factor. Division and Factors continued • Divide: x2 x 1 x 4 x3 2 x 2 5 x 4 x3 x 2 x2 5x x2 x 4x 4 4x 4 0 remainder Since the remainder is 0, we know that x 1 is a factor. How do you divide a polynomial by another polynomial? • Perform long division, as you do with numbers! Remember, division is repeated subtraction, so each time you have a new term, you must SUBTRACT it from the previous term. • Work from left to right, starting with the highest degree term. • Just as with numbers, there may be a remainder left. The divisor may not go into the dividend evenly. The Remainder Theorem If a number c is substituted for x in a polynomial f(x), then the result f(c) is the remainder that would be obtained by dividing f(x) by x c. That is, if f(x) = (x c) • Q(x) + R, then f(c) = R. Synthetic division is a “collapsed” version of long division; only the coefficients of the terms are written. Synthetic division is a quick form of long division for polynomials where the divisor has form x - c. In synthetic division the variables are not written, only the essential part of the long division. x x 2 x 2 3 2 1 5 6 x 2 x 5x 6 2 x 2x -2 -6 _______ 1 3 0 3x 6 quotient 3x 6 _______ 2 0 remainder x 5x 6 ( x 2)( x 3) 2 Example Use synthetic division to find the quotient and remainder. 5 4 3 2 4 x x 6 x 2 x 50 ( x 2) 2 –4 1 6 2 0 50 –8 –14 –16 –28 –56 –4 –7 –8 –14 –28 –6 Note: We must write a 0 for the missing term. The quotient is – 4x4 – 7x3 – 8x2 – 14x – 28 and the remainder is –6. Example continued: written in the form P ( x) d ( x) Q( x) R( x) 5 4 3 2 4 x x 6 x 2 x 50 x 2 -4x 4 7 x 8 x 14 x 28 6 3 2 By the remainder theorem we know P(2) 6 Example • Determine whether 4 is a zero of f(x), where f(x) = x3 6x2 + 11x 6. • We use synthetic division and the remainder theorem to find f(4). 4 1 –6 11 –6 4 –8 12 1 –2 3 6 • Since f(4) 0, the number is not a zero of f(x). The Factor Theorem • For a polynomial f(x), if f(c) = 0, then x c is a factor of f(x). Example: Let f(x) = x3 7x + 6. Solve the equation f(x) = 0 given that x = 1 is a zero. Solution: Since x = 1 is a zero, divide synthetically by 1. 1 1 0 -7 6 1 1 -6 1 1 -6 0 Since f(1) = 0, we know that x 1 is one factor and the quotient x2 + x 6 is another. So, f(x) = (x 1)(x + 3)(x 2). For f(x) = 0, x = 3, 1, 2. Factor Theorem • f(x) is a polynomial, therefore f(c) = 0 if and only if x – c is a factor of f(x). • If we know a factor, we know a zero! • If we know a zero, we know a factor! • Definition of Depressed Polynomial • A Depressed Polynomial is the quotient that we get when a polynomial is divided by one of its binomial factors • Which of the following can be divided by the binomial factor (x - 1) to give a depressed polynomial (x - 1)? Choices: A. x2 - 2x + 1 B. x2 - 2x - 2 C. x2 - 3x - 3 D. x2 - 2 Using the remainder theorem to find missing coeffecients… • Find the value of k that results in a remainder of “0” given… ( x 3x kx 24) ( x 3) 3 2