Solving+Polynomial+Equations

= **Solving Polynomial Equations ** =

Wednesday, October 5th, 2011
Example Question: Solve for //x//: 3//x//3 + 10//x//2 – 7//x// = //x//3+3//x//2+30

First: Arrange so everything is equal to zero: 3//x//3 –//x//3 +10//x//2 – 3//x//2 -7//x// – 30 = 0  2//x//3 + 7//x//2 – 7//x// – 30 = 0

To find a zero, substitute a number for //x//, to see if the equation will equal zero. We tried using 2 in class: 2(2)3 + 7(2)2 – 7(2) – 30 = 0  8 + 28 – 14 – 30 = 0  0 = 0 Therefore, //f//(//2//) = 0. Use synthetic division to find the next formula, and since the remaining formula will be quadratic, factor the equation to find the remaining two zeroes.
 * 2 || 2  || 7  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">-7  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">-30  ||
 * ||  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">4  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">22  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">30  ||
 * || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">2 || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">11  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">15  || <span style="display: block; font-family: Garamond,serif; font-size: 12pt; text-align: center;">0  ||

<span style="font-family: Garamond,serif; font-size: 12pt;"> Continue with the new equation, and factor: <span style="font-family: Garamond,serif; font-size: 12pt;">(//x// – 2)(2//x//2 + 11//x// + 15) = 0 <span style="font-family: Garamond,serif; font-size: 12pt;"> (//x// – 2)[(2//x//2 + 6//x// + 5//x// + 15) = 0 <span style="font-family: Garamond,serif; font-size: 12pt;"> (//x// – 2)[2//x//(//x// + 3) + 5(//x// + 3)] = 0 <span style="font-family: Garamond,serif; font-size: 12pt;"> (//x// – 2) (2//x// + 5) (//x// + 3) = 0

<span style="font-family: Garamond,serif; font-size: 12pt;">Therefore, //x// = 1, -5/2 and -3 <span style="font-family: Garamond,serif; font-size: 12pt;">Sometimes, you may be asked to check: <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 3(8) + 10(4) – 14 <span style="font-family: Garamond,serif; font-size: 12pt;"> LS =50 || <span style="font-family: Garamond,serif; font-size: 12pt;">RS = (2)3+3(2)2+30 <span style="font-family: Garamond,serif; font-size: 12pt;">RS =8 + 12 + 30 <span style="font-family: Garamond,serif; font-size: 12pt;">RS = 50 || <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 3(-125/8) + 10 (-25/4) + 35/2 <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 265/8 || <span style="font-family: Garamond,serif; font-size: 12pt;">RS = (-5/2)3+3(-5/2)2+30 <span style="font-family: Garamond,serif; font-size: 12pt;">RS = -125/8 + 150/8 + 240/8 <span style="font-family: Garamond,serif; font-size: 12pt;">RS = 265/8 || <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 3(-27) + 10 (9) + 21 <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 30 || <span style="font-family: Garamond,serif; font-size: 12pt;">RS = (-3)3+3(-3)2+30 <span style="font-family: Garamond,serif; font-size: 12pt;">RS = -27 + 27 + 30 <span style="font-family: Garamond,serif; font-size: 12pt;">RS = 30 ||
 * <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 3(2)3 + 10(2)2 – 7(2)
 * <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 3(-5/2)3 + 10(-5/2)2 – 7(-5/2)
 * <span style="font-family: Garamond,serif; font-size: 12pt;">LS = 3(-3)3 + 10(-3)2 – 7(-3)