Monday, January 9, 2017

Sum and Difference of Cubes

We are about four days into our polynomials unit (Algebra 2 Honors) and want students to see that being given a polynomial in standard form is simply not as useful as having it in factored form. So, we're going to spend a day practicing factoring.

That said, they've never seen how to factor the sum and difference of cubes. And we didn't want to just TELL them. So, instead, we thought of a way to have them figure it out!

Follow the link here for the whole handout, titled "4. Factoring"

Overall, I think it's helping students to make connections! Thought I would share the resource!

1.     Given 4 is an x-intercept of f(x), what must be a factor of f(x)?

2.     A portion of the graph of a polynomial is below, what (if anything) do you know about…

a) the degree of the polynomial?

b) its end behavior?

c) its roots? Their multiplicity?

3.     Identify the end behavior and x-intercepts of

a.  h(x)=x(x+3)^3(2-x)                                             b)g(x)=-x^5 +4x^4-4x^3        

c)  f(x)=x^5-3x^4-x^3+3x^2

Almost all the polynomials we have explored so far have been in factored form, which is convenient as we can easily find the intercepts.  Sadly, polynomials are often in standard form, which is far less convenient.  We already know to take a quadratic (a second degree polynomials) from standard from to factored form (by factoring).  This worksheet will serve as a review of those methods as we all exploring a few other ways of factoring as well. 

Sum/ Difference of Cubes

1.     On Desmos, or your graphing calculator, graph y = x^3 + 8 
           a.  What is the x-intercept? What is its multiplicity? 

            b. Given your answer above, what must be a factor of x^3 + 8? 

This means that (x^3 + 8)=(answer from above)(something).  But how do we find that something?

            c.  What degree does something have to be? 

Therefore (x^3 + 8)=(answer from above)(ax^2 + bx +c). 

            d.  Using what you know about distribution, what does a have to be?  c? 

            e.  Find b. 

            f.  So factored, x^3 + 8 = (             ) (                               ). 

2.     Try to same process to factor, x^3 - 27 

 (this is on the next page)
What you hopefully found is that
(insert answer to first)  and (answer to second)        (check your answers)

These are categorized as the sum and difference of cubes.  And are factorable as

            Give formulas here!  

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