6/23 HW and Lesson Overview

HW#25: p. 218 E.E. 11-17 ODD, W.E. 9-19 ODD

Recall the zero product property: If ab=0, then either a=0 or b=0 (or both can equal 0).  So for example, if (x-3)x=0, either x-3=0, or x=0.  Solving for x in the first equation gives x=3, and then x=0 was the other option, so the answer is x=3 or 0.

Here, for E.E., the problems are already factored!  Do not multiply them out!  Factored form is supposed to be convenient and easy to work with due to the zero product property, so use that!

For W.E., factor the trinomials, and then use the zero product property.

HW#26 p. 312-313 W.E. 1, 5, 9-23 ODD

p. 174 E.E. 1-4

Determine what “a” is.  Remember, synthetic division is for dividing f(x) by x-a, not just by a.  Example: To divide a polynomial by x+2 using synthetic division, a=-2

The remainder theorem then tells you, what ever the remainder is after dividing f(x) by x-a, is equal to f(a).  Remember, f(a) means the value of the function f(x) when x=a.  It’s like taking all of the x’s in the polynomial and replacing them with a’s.  What the remainder theorem does, is give us a short cut.  Once more: f(a) = the remainder from f(x)÷(x-a).

HW#27: p. 230 W.E. 23-35 Col 2

Recall: if (x+2)(x-3) = 0, then x=-2 or 3 are the solutions, roots, or zeros of the equation.  Going the other way, if I know that x=4, or -3/2 are the roots, zeros, or solutions to some equation, I can use the roots to determine the factored form of the original polynomial.

If x=4, then x-4=0, so (x-4) is a factor of the polynomial.

If x=-3/2, then 2x=-3, and 2x+3=0, so (2x+3) is the other factor of the polynomial.

Since (2x+3) and (x-4) are the factors of the original polynomial, (2x+3)•(x-4) must give the original polynomial.  Multiply it out to get f(x)=2x^2-5x-12, or y=2x^2-5x-12

A.) Mike is trying to sketch a graph of the polynomial

f(x)=x^3+4x^2+x−6.

He notices that the coefficients of f(x) add up to zero (1+4+1−6=0 ) and says

This means that 1 is a zero of f(x), and I can use this to help factor f(x) and produce the graph.

  1. Is Mike right that 1 is a zero of f(x)? Explain his reasoning.
  2. Find all zeros of f(x).

B.) Consider the polynomial function

p(x)=x^4−3x^3+ax^2−6x+14,

where a is an unknown real number. If (x−2) is a factor of this polynomial, what is the value of a?

Hint: Set up a box or do synthetic division.  What does it mean for something to be a factor?  Work forwards until you reach the variable, then work backwards from the end to reach the variable from the other way.

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