Today's soundtrack is S.O.: So it Begins, a criminally underrated rap album. I love everything about it: the reformed lyricism, the skillful rapping, the lush samples.
Yesterday, I started learning about the Remainder Theorem. Today, I'm going to work on the assignment portion of the lesson and go through some examples of the different exercises.
How to determine whether a polynomial is a factor of another polynomial
Set up the equation
(x³ - 7x + 6) / (x + 3)
Find the value of the dividend as a root
x + 3 = 0; x = -3
Substitute the value into the equation
P(-3) = -3³ - 7(-3) + 6
Simplify
P(-3) = -27 + 21 + 6
P(-3) = -27 + 27
P(-3) = 0
If there is no remainder, the dividend is a factor of the first polynomial. If there is a remainder, the dividend is not a factor.
How to determine the value of a variable, given the equation and the remainder
Survey the information provided
If given (x² + mx - 7) / (x - 2), remainder = 3, we know that we are going to need to solve for m.
Put the remainder on the left side, substitute the root value of the constant in for x
3 = 2² + 2m - 7
Simplify
3 = 4 + 2m - 7
3 = -3 + 2m
Isolate the variable
6 = 2m
Divide both sides by the variable's coefficient
6/2 = 2m/2
3 = m
m = 3
Here's one more example of the same, but a bit more complex:
Survey the information provided
If given (x⁴ - 3x³ + x² + mx + 5) / (2x - 1), remainder = 47/16, we know that we are going to need to solve for m. Solving 2x-1 for x, we see that 2x-1=0 is 2x=1; dividing 2x=1 by 2 gives us x = 1/2, so we will substitute 1/2 in for x in the equation.
Set up the equation
47/16 = (1/2)⁴ - 3(1/2)³ + (1/2)² + m(1/2) + 5
Simplify
47/16 = 1/16 - 3(1/8) + 1/4 + 1/2m + 5
47/16 = 1/16 - 6/16 + 4/16 + 8/16m + 80/16
Isolate the variable
-8/16m = -47/16 + 79/16
-8/16m = 32/16
-1/2m = 2
Divide both sides by the variable's coefficient
(-1/2m = 2) / (-1/2)
m = 4
How to write a division statement after solving a question
Solve the problem using either polynomial long division or synthetic division
(x³ - 2x² - 15x - 17) / (x + 2)
We carry down 1, -2 ⋅ 1 = -2, -2 - 2 = -4, -4 ⋅ -2 = 8, -15 + 8 = -7, -7 ⋅ -2 = 14, -17 + 14 = 3. We get x²-4x-7, remainder 3.
Using the format DIVIDEND = (DIVISOR) (QUOTIENT) + REMAINDER , input the answer
(x³ - 2x² - 15x - 17) = (x² - 4x + 7) (x + 2) - 3