Dividing Polynomials: Remainder and Factor Theorems


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1 Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend. Quotient Divisor Dividend The how to of Long Division 1. Arrange the terms of both the dividend and the divisor in decreasing exponential order 2. Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient 3. Multiply the every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. 4. Subtract the product from the dividend 5. Bring down the next term in the original dividend and write it next to the remainder to form a new dividend 6. Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. This will occur when the degree of the remainder is less than the degree of the divisor. The Division Algorithm When we divide a polynomial, P(x), by a divisor, d(x), a polynomial Q(x) is the quotient and a polynomial R(x) is the remainder. P(x) = d(x) Q(x) + R(x) 1 Page
2 For example P(x) = x 3 + 2x 2 5x 6 and d(x) = x + 1 then we will see in the next example that Q(x) = x 2 + x 6 and R(x) = 0 such that P(x) = d(x) Q(x) + R(x) can now be written P(x) = (x + 1) (x 2 + x 6) + 0 Example Divide to determine whether x + 1 is a factor of x 3 + 2x 2 5x 6 2 Page
3 Example Divide 6x 4 + 5x 3 + 3x 5 by 3x 2 2x. Is the divisor a factor of the dividend? Remainder Theorem If a number c is substituted for x in the polynomial f(x) then the result f(c) is the remainder that would be obtained by dividing f(x) by x  c. That is f(x) = (x c) Q(x) + R, then f(c) = R 3 Page
4 For example consider h(x) = x 3 + 2x 2 5x 6. When we divided h(x) by x + 1 and x 3, the remainders where 0 and 24. What are the function values of h( 1) and h(3). 4 Page
5 The how to of Synthetic Division 1. Arrange the polynomial in descending exponential order, with a 0 coefficient for any missing terms. 2. Write c for the divisor, x c. To the right, write the coefficients of the dividend. 3. Write the leading coefficients of the dividend on the bottom row. 4. Multiply c times the value just written on the bottom row. Write the product in the next column in the second row. 5. Add the values in this new column, writing the sum in the bottom row. 6. Repeat this series of multiplications and additions until all columns are filled in. Example Divide 4x 3 3x 2 + x + 7 by x 2 is the divisor a factor of the dividend? If the remainder is 0 then answer No. C = Example Divide 2x 3 + 7x 2 5 by x + 3 is the divisor a factor of the dividend? If the remainder is 0 then answer No. C = 5 Page
6 Example Determine whether 4 is a zero of f(x)? f(x) = x 3 + 8x 2 + 8x 32 C = Example Determine whether i is a zero of f(x)? f(x) = x 3 3x 2 + x 3 C = The Factor Theorem Let f(x) be a polynomial f(x) = (x c)q(x) + r a) If f(c) = 0, then x c is a factor of f(x) b) If x c is a factor of f(x), then f(c) = 0 6 Page
7 Example Solve the equation 2x 3 3x 2 11x + 6 = 0 given that 3 is a zero of f(x) = 2x 3 3x 2 11x + 6 Example Solve the equation 15x x 2 3x 2 = 0 given that 1 is a zero of f(x) = 15x x 2 3x 2 7 Page
8 Zeros of Polynomial Function Recall: Natural or Counting Numbers (N) {1,2,3,4, } Whole Numbers (W) {0,1,2,3,4, } Integers (Z) {,3,2,1,0,1,2,3, } Rational Numbers (Q) p p and q are integers, and q 0} q Real Numbers (R) {x x is a number that can be expressed as a decimal} 3 Irrational Numbers Examples { 2, 4, and π} {x x is a real number and x cannot be expressed as a quocient of integers} Z W R N Q The Rational Zero Theorem If f(x) = a n x n + a n 1 x n a 1 x + a 0 has integer coefficients and p (where p is reduced to lowest terms) is a rational zero of f, q q then p is a factor of the constant term a 0 and q is a factor of the leading coefficient, a n. 8 Page
9 Example List all possible rational zeros of f(x) = 3x 4 11x x 4 Example  List all possible rational zeros of f(x) = x 4 + 3x and find the zeros of f(x). 9 Page
10 Example Solve: f(x) = x 4 6x 2 8x + 24 = 0 Properties or Roots of Polynomial Equations 1. If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. 2. If a + bi is a root of a polynomial equation with real coefficients (b 0), then the imaginary number a bi is also a root. Imaginary roots, if they exist occur in conjugate pairs. For example if 2 + 7i is a zero of a polynomial function f(x) with real coefficients, then the conjugate 2 7i is also a zero. 10 Page
11 The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n 1, then the equation f(x) = 0 has at least one complex root (that is at least one zero in the system of complex numbers). Note: that complex numbers include both real numbers b = 0 and imaginary numbers b 0 The Linear Factorization Theorem If f(x) = a n x n + a n 1 x n a 1 x + a 0 where n 1 and a n 0, then f(x) = a n (x c 1 )(x c 2 ) (x c n ) where c 1, c 2, c n are complex numbers. In words: An nth degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1 Example Find a polynomial function of degree 3 having zeros 1,3i and 3i C 1 = C 2 = C 3 = 11 Page
12 Example Find a polynomial function of degree 5 with 1 as a zero of multiplicity 3, 4 as a zero of multiplicity 1, and 0 a zero of multiplicity 1 C 1 = C 2 = C 3 = C 4 = C 5 = Example Find a fourth degree polynomial function with real coefficients that has 2, 2 and i as zeros and such that f(3) = 150. C 1 = C 2 = C 3 = C 4 = 12 Page
13 Descartes s Rule of Signs Let P(x), written in decreasing exponential order or ascending exponential order be a polynomial function with real coefficients and a nonzero constant term. The number of positive real zeros of P(x) is either: 1. the same as the number of variations of sign in P(x) 2. Less than the number of sign variations of sign in P(x), by a positive even integer. If P(x) has only one variation in sign then P has exactly one positive real zero. The number of negative real zeros of P(x) is either: 1. the same as the number of variations of sign in P(x) 2. Less than the number of sign variations of sign in P(x), by a positive even integer. If P(x) has only one variation in sign then P has exactly one negaitve real zero. 13 Page
14 Example Determine the possible number of positive and negative real zeros of P(x) = x 3 + 2x 2 + 5x + 4 Step 1: P(x) = How many sign changes? Positive Real Zeros Step 2: P( x) = How many sign changes? Negative Real Zeros 14 Page
15 Example Determine the possible number of positive and negative real zeros of P(x) = 5x 4 3x 3 + 7x 2 12x + 4 Step 1: P(x) = How many sign changes? Positive Real Zeros Step 2: P( x) = How many sign changes? Negative Real Zeros 15 Page
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