factoring x^3 - 6x^2 + 10x - 8 (how to explain to students)
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Textbooks usually cover this material in a fairly sensible order. Try using the techniques which recently preceded this lesson, along with what you learned back when you took algebra. Apply the Rational Roots Test, run the potential zeroes through synthetic division (using Descartes' Law, as needed), etc, etc. You can also try following one of the worked examples in the book, expanding the text's explanation as you feel necessary.pamela said:
At what point in your lecture do your students appear to be getting lost?
Eliz.
Re: factoring
For your students, or for you? :?:
\(\displaystyle x^3 - 6x^2 +10x - 8\)
Read what Eliz said.. it is important that both you and your students understand all the methods that can be used :!:
We can do some simple manipulation of terms.
Just by looking at the answer, we can reconstruct the terms prior to grouping & factoring. (If we didn't have the answer, it would still be doable, it would just take a little longer to see what needed to be manipulate
\(\displaystyle x^3 - 2x^2 + 2x - 4x^2 + 8x - 8\)(if you simplify these terms, you will get \(\displaystyle x^3 - 6x^2 +10x - 8\)
\(\displaystyle x(x^2 - 2x + 2) - 4(x^2 - 2x + 2)\)
\(\displaystyle = (x - 4)(x^2 - 2x + 2)\)
This one, of course, is a little tricky..... it would be easier to use the rational roots test and then synthetic division instead of playing around with terms for 20 minutes
Descartes' Law? Isn't that related to optics? :shock:stapel said:
pamela said:
For your students, or for you? :?:
\(\displaystyle x^3 - 6x^2 +10x - 8\)
Read what Eliz said.. it is important that both you and your students understand all the methods that can be used :!:
We can do some simple manipulation of terms.
Just by looking at the answer, we can reconstruct the terms prior to grouping & factoring. (If we didn't have the answer, it would still be doable, it would just take a little longer to see what needed to be manipulate
\(\displaystyle x^3 - 2x^2 + 2x - 4x^2 + 8x - 8\)(if you simplify these terms, you will get \(\displaystyle x^3 - 6x^2 +10x - 8\)
\(\displaystyle x(x^2 - 2x + 2) - 4(x^2 - 2x + 2)\)
\(\displaystyle = (x - 4)(x^2 - 2x + 2)\)
This one, of course, is a little tricky..... it would be easier to use the rational roots test and then synthetic division instead of playing around with terms for 20 minutes
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pamela said:
You might want to start first by reviewing the Rational Roots Theorem, which will tell you what all possible rational roots are.
If you find ONE rational root (and you'll recognize it because the value of the polynomial will be 0 when you substitute that root for x), you can use division (either long division or synthetic division) to get a "reduced polynomial" which may or may not be further factorable.
D
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My approach would be to use technology - graph it!!pamela said:
That gives you a good idea about whether you have any rational root or not.