As you probably remember, last year we learnt to solve sytems of linear equations (first degree equations). Now we are going to try something a bit more difficult: systems of nolinear equations. In these systems, one or more of the equations are at least of second degree and/or there is a product of variables in one of the equations.. The method we are going to use is substitution, because it’s the easiest, and in many cases the only practical way to solve these problems. Sometimes elimination works as well, or equalization. The graphic method is very helpful here, if you have access to geogebra, for instance. (HINT: I bet you have, it works on your phone! →Download page)

Now you have several tools to factorize polynomials (common factor, Ruffini, second degree formula).

For example, a second degree polynomial can be factorized using the formula, and writing it this way:

a x^{2} + bx + c =a · (x − s_{1}) · (x − s_{2})

being s1, s2, the solutions or roots of the polynomial.

An important word to use here is root. We say that x=a is a root or zero of the polynomial P(x) if P(a)=0. That means that this same x=a is a solution of the equation P(x)=0

With that in mind, you can use the method we saw for biquadratic equations as well. Use it to find the solutions (=roots) and then you can write the factorization this way:

ax^{4} + bx^{2} + c =a·(x-s1)(x-s2)(x-s3)(x-s4)

being s1, s2, s3, s4 the solutions or roots of the polynomial,

Now some exercises to work out in your notebook. Use Symbolab to check them.

Factor and Calculate the Roots of the Following Polynomials

a) x^{3 } + x^{2 }

b) 2x^{4} + 4x^{2}

c) x^{2} − 4

d) x^{4} − 16

e) 9 + 6x + x^{2}

f)

g) x^{4} − 10x^{2} + 9

h) x^{4} − 2x^{2} − 3

i) 2x^{4} + x^{3} − 8x^{2} − x + 6

Now solve these equations. You should use the same tools, the only difference is the solution you have to give me.

That’s it. The Zero Product Property simply states that if ab=0 , then either a=0 or b=0 (or both). A product of factors is zero if and only if one or more of the factors is zero. Pretty logical, huh?

As simple as it seems, it’s so powerful that we can solve polynomial equations of any degree using this method, if we can factor the polynomial (using Ruffini, for example).

Here, we have found, for example, that 1 and -2 are solutions of

There’s one solution more that we could find with Ruffini. Can you find it?

It also explains the relationship between factors of a polynomial and x-intercept of a function:

For example, if we plot this polynomial as a function:

We get this:

And you can see that the function crosses the x-axis on x=1 and x=-2 (and in x=-1, because we could also get this with ruffini)

All in all, it´s clear that zero is a very interesting number (although very feared by students all over the world ).

A very complete (and lenghty) summary of this can be found here.

1. First of all, 5 minutes to check the equations from yesterday using this page of Symbolab.

2. Now solve as many of these equations as you can in a blank sheet of paper (to give to the teacher at the end of the class, with your name on it) and check the solutions with Symbolab.

a) x^{4} − 10x^{2} + 9 = 0

b)

c) x^{4} − 61x^{2} + 900 = 0

d)

e)

f)

g) Two natural numbers differ by two units and the sum of their squares is 580. What are these numbers?

h) A rectangular garden 50 m long and 34 m wide is surrounded by a uniform dirt road. Find the width of the road if the total area of the garden and road is 540 m².

We are going to practice a little bit about division of polynomials, including Ruffini’s rule.

1. The teacher is going to show you some tools to correct these type of exercises. Pay attention. (NOTE: The programs use to show the answer in this format: D/d=Q+R/d)

2. Use these tools to correct the exercises from this week, at least the products and divisions.

3. Work out these divisions (in your notebook) using ruffini’s rule:

5. Use the tools mentioned above to correct them.

6. Send me an email with the answers.

7. Now do the same thing with this exercise from your book: page 49, ex. 7.

8. Last thing to do: Find out who Paolo Ruffini was, and send me a few notes about him.

0. First thing to do, a little bit of mental calculation: “10 seconds of math”, five minutes, but this time click squares and roots as well.

2. We are going to insist in the rationalizations of fractions. Go to this page and try to understand the two (some say three) different cases. Solve the questions at the end (only 1, 2 and 3).

0. First thing to do: a little history project. Search in the internet to find when and by whom the different sets of numbers were “discovered”. Write the information in a word document and send it to me via email. You have 20 minutes for this task.

1. Some reading about roots. Solve the exercises of the end of the page in your notebook.

3. Well, it’s time for a new thing, and a bit difficult. We are going to learn to rationalize a fraction. Go to this page and try to understand the two (some say three) different cases. Copy in your notebook the theory and try to solve the questions.