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03.17.2019: One Lesson of Math - Polynomials, 1/5: The Remainder Theorem, Part 2


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

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