Synthetic Division/Rational Roots -- Section 5.3 -- Use synthetic division to determine if (x - 1) is a factor of  P(x) = x3 - 2x2 - 5x + 6
 Put 1 in the upper left hand corner (box). Put the coefficients of the polynomial in the first row. Solve Step Bring the down the 1. Use  1 (in box) if  x - 1 is the factor. For x - r , use r Multiply 1 × 1 = 1 Add Columns -2 + 1 = -1 Multiply -1 × 1 = 1 Add Columns -5 + (-1) = -6 Multiply -6 × 1 = -6 Add Columns 6 + (-6) = 0 If there is a remainder of zero, 0 , then 1 is a zero and (x -1) is a factor of the polynomial.  Use synthetic division to determine if (x - 1) is a factor of  P(x) = 2x3 - 2x2 - 5x + 6 [Solution]  Find all rational roots of  P(x) = 2x3 + 3x2 - 14x - 15

 Find all possible rational zeros - rational numbers are ratio of integers i.e. -2/1 = -2, 0, 1/2, -3/2 Solve Step p = ± 1,  ± 3,  ± 5,  ± 15 q ± 1, ± 2 Find possible rational zeros p/q where p are factors of the last coefficient 15.  q are the factors of the first coefficient 2. p = ± 1,  ± 2,  ± 3,  ± 6  and p/1 = p  and all of the p/2's. We always get the p's Try the easy ones first. Start with -1 (I tried 1 first but it failed - You can use Descartes' Rules of Signs to narrow the search.) Put -1 in the upper left hand corner (box). Put the coefficients of the polynomial in the first row. Bring the down the 2. coefficient of 2x3 is 2 Multiply 2 × -1 = -2 Add Columns 3 + (-2) = 1 Multiply 1 × -1 = -1Add Columns -14 + (-1) = -15 Multiply -15 × -1 = 15 Add Columns -15 + 15 = 0 If there is a remainder of zero, 0 , then -1 is a zero and (x -(-1)) = (x + 1) is a factor of the polynomial. Q(x) = 2x2 + x - 15 The quotient is the coefficients of the bottom line Once we have a quadratic equation, factor or use the quadratic formula. 2x2 + x - 15 = (2x - 1)(x + 3) = 0 Set equal to zero and solve for x 2x - 1 = 0   or   x + 3 = 0 x = 1/2   or x = -3 Check in original equation. { -1, 1/2, -3} There are three solutions for P(x) P(1/2) = P(-1) = P(-3) = 0 Side note:  P(x) = 2x3 + 3x2 - 14x - 15 = (x + 1)(2x - 1)(x + 3)  Find all possible rational roots and all rational roots of P(x) = 4x3 - 15x2 - 31x + 30 [Solution]  Find all roots of  P(x) = 3x4 + 10x3 - 9x2 - 40x - 12
 Find all possible rational zeros - rational numbers are ratio of integers i.e. -2/1 = -2, 0, 1/2, -3/2 Solve Step Property p = ± 1,  ± 2,  ± 3,  ± 4, ± 6,  ± 12 q ± 1, ± 3 Find possible rational zeros p/q where p are factors of the last coefficient 12.  q are the factors of the first coefficient 2. p = ± 1,  ± 2,  ± 3,  ± 4, ± 6,  ± 12 and p/1 = p We always get the p's and all of the p/3's. Reduce Start with 2 (I tried others first - You can use Descartes' Rules of Signs to narrow the search.) Bring the down the 3. coefficient of 3x4 is 3 Multiply 3 × 2 = 6Add Columns 10 + 6 = 16 Multiply 16 × 2 = 32Add Columns -9 + 32 = 23 Multiply 23 × 2 = 46Add Columns -40 + 46 =  6 Multiply 6 × 2 = 12Add Columns -12 + 12 = 0 If there is a remainder of zero, 0 , then 2 is a zero and (x - 2)  is a factor of the polynomial. Q(x) = 3x3 + 16x2 + 23x + 6 The quotient is the coefficients of the bottom line Try -2 next I tried other numbers that failed first I did the work for you.... If there is a remainder of zero, 0 , then 2 is a zero and (x - (-2)) = (x + 2) is a factor of the polynomial. Q(x) = 3x2 + 10x + 3 The quotient is the coefficients of the bottom line Once we have a quadratic equation, factor or use the quadratic formula. 3x2 + 10x + 35 = (3x + 1)(x + 3) = 0 Set equal to zero and solve for x 3x + 1 = 0   or   x + 3 = 0 x = -1/3   or x = -3 Check in original equation. { 2, -2, -1/3  , -3} There are three solutions for P(x) P(2) = P(-2) = P(-1/3) = P(-3) = 0 Side note:   P(x) = 3x4 + 10x3 - 9x2 - 40x - 12 = (x - 2)(x + 2)(3x + 1)(x + 3) Tutorials and Applets by
Joe McDonald