Synthetic Division By: Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z.
What is Synthetic Division? Synthetic Division is a simpler way to divide a polynomial by a linear factor. You can consider it a shorter way compared to long division. By finding the linear factors, you can find the zeros/roots of the polynomial. With the zeros/roots, you are able to graph the polynomial easily.
Key Terms Synthetic Substitution- When synthetic division is used to evaluate a function. S.Z. Rational Roots Theorem- This helps you to find the possible roots of a polynomial. (see example 1) Descartes' Rule of Signs- This helps you to predict the number of positive and negative real roots of a polynomial. (see example 1) v.c.
Example #1 x 3-3x 2-4x +12 x + 2 rational roots theorem: (note: the factors are + ) 12: +1 +2 +3 +4 +6 +12 1: (<---this came from the coefficient of x 3 ) +1 possible roots: +1 +2 +3 +4 +6 +12 (they can also be fractions depending on the factors of the highest term's coefficient.) Descartes' rule: x 3-3x 2-4x +12 + - - + (<--these signs are the positive/negative signs of the coefficient of each term.) ^ ^ ^ yes no yes (<-- this refers to whether or not if there is a sign change.) (continues on next slide...) v.c
Example #1 + - i 2 1 0 0 1 2 (the i means any possible imaginary numbers.) > for both, the total number of roots, should be three since there should only be three roots. ^ (the positive values go down by two! until it reaches 0) synthetic division: -2 1-3 -4 12 (The last number under the line must equal zero for the value you -2_10_-12 chose to work.) 1-5 6 0 new equation = x 2-5x + 6 linear factor: x + 2 ^ the "x+2" comes from the synthetic division "-2" *since a negative root has been found, there should be two positive roots since it seems unlikely that there would be imaginary numbers in the rest of this polynomial. (continued on next slide...) v.c.
Example #1 2 1-5 6 2_-6 1-3 0 x - 2 = 0 new equation: x - 3 linear factors: x - 2 and x + 2 + 2 +2 x = 2 OR (depending whichever value you chose!) 3 1-5 6 3_-6 1-2 0 x - 3 = 0 new equation: x - 2 linear factors: x - 3 and x + 2 +3 +3 x = 3 The linear factors are x+2, x-2, and x-3. v.c.
Using Synthetic Division to GRAPH Example #2 2x 3 + 7x 2-16x + 6 - Through the Rational Root Theorem, the possibilities of the zeroes could be.. + 1, 2, 3, 6, 1/2, 3/2 - The addition of the exponents which is 5, is how many x-intercepts there could be Through the Graph, we could see that the x-intercepts aka the zeros are 1/2 and one seems as if it's a little before -5 while the other one is a little after 1. M.R.
- The polynomial graph could change directions if it's either a positive or a negative. Graphing Facts ; - Looking at the exponents of the polynomials could also tell how the graph is going to look like, with loops or a straight line, etc. - When using synthetic division, you find out the factors which for example usually could look like -> (x-3) so, that makes -3 a root of that polynomial. f(x) = ax n +... +ax + a Fact - At most there could be n, zeros, factors and x-intercepts and (n-1) turning points which could also be the humps. Example - 1. A polynomial of the degree of 7, could have 7 zeros, 7 factors, 7 x-intercepts and 6 turning points aka humps. When the factors of the polynomial don't have a degree, it means its 1. Any factor with the degree of one, makes the straight line through that root on a graph. But if the degree is 2 that on the graph, there is U and the bottom of the U which would be the vertex is the x-intercept. And if the degree is cubic then the line on the graph has a straight line and when it touches that x-intercept it flattens, then again has a straight line again.
Graphing Example Practice - x 4-13x 2 + 36 Factoring - (x 2-9)(x 2-4) (x 2-9) = (x + 3)(x - 3) (x 2-4) = (x + 2)(x - 2) so the roots would be, 2, -2, 3, -3. M.R.
FACTORS Example #3: FACTORING: Finding if x - 4 is a factor of: -2x 5 + 6x 4 + 10x 3-6x 2-9x + 4 In order to test the factor we must make it into a zero so instead of x - 4 the factor becomes the zero, x = 4. 4-2 6 10-6 -9 4-8 -8 8 8-4 -2-2 2 2-1 0 Now because the remainder is 0 we can conclude that x = 4 is a zero of the equation. Therefore x - 4 is a factor of -2x 5 + 6x 4 + 10x 3-6x 2-9x + 4. P.S.
ROOTS Example #4: To find the roots of an equation we would use the Rational Zeros Theorem. -2x 4 +x 3-19x 2-9x + 9 Factors of leading coefficient: +/-1, +/-2 Factors of constant term: +/-1, +/-3, +/-9 Possible Values of p/q: +/- 1/1, +/- 1/2, +/- 3/1, +/- 3/2, +/- 9/1, +/- 9/2 Simplified Possible Values: +/- 1, +/- 1/2, +/-3. +/- 3/2, +/- 9, +/- 9/2 (i.e - Non-Zero) (i.e. - Zero) 1 2 1-19 -9 9 1/2 2 1-19 -9 9 2 3-16 -25 1 1-9 -9 2 3-16 -25-16 (b/c this does not end in 0, 1 is not a root) 2 2-18 -18 0 (1/2 is indeed a root b/c the result is 0) P.S.
BE CAREFUL! Don't get confused! Here are some common errors that people might make when they are using Synthetic Division: ~ When you see the divisor of the problem, or the linear factor that you will be dividing the polynomial into, remember to switch the sign of the original number when you put it into synthetic division form. ^ Example #5 x - 3 ) x + x -14 3 1 1-14 (continued on next slide...) S.Z.
Continued... ~ When you begin solving the synthetic division problem, remember that you MULTIPLY the number on the left side of the synthetic division sign with the consecutive number on the bottom of the sign and you ADD the numbers that are underneath each other, inside the synthetic division sign. ^ Example #5 continued... 3 1 1-14 (3 x 1 = 3) 3 1 4 (1 + 3 = 4)
Practice Problems 1. f(x) = 2x 5-3x 4 + 2x 2 + x - 8 2. g(x) = 5x 5-4x 4 + 19x 3-16x 2 - x - 4 3. p(x) = x 6 + x 5-13x 2 - x + 12 4. h(x) = -2x 4-14x 3 + 71x 2-173x + 150 S.Z.