I have to warn you that the formatting is not as nice as if you were looking at a regular Word document--the post editor won't let me cut-and-paste from a Word document, and I couldn't find a way to type nice exponents.
Here's the best I could do:
"x squared" looks like "x^2"
"plus or minus" looks like +/-
"square root" looks like "sqrt"
Leave your name and a comment if you have a question, and I will reply by leaving a comment.
1A. x^5 - x^3 + x --> quintic trinomial
1B. 2x^2 + 2x --> quadratic binomial
2A. y = -.04x^3 + .65x^2 - 2.94x + 24.84 --> y=7.33 when x = 12
3A. y = x^3 - x^2 -25x + 25
3B. y = x^3 + 6x^2 + 12x + 8
4A. 3x^2 (x - 2)(x + 1)
4B. 4x (x + 5)(x - 2)
5A. x = -1 --> multiplicity 2 AND x = 3
5B. x = 0 & x = 2 -->multiplicity 3 AND x = -5
6A. x^3 + 3x^2 + 3x +4, R1
6B. 3x^2 - 7x + 7, R-8
7A. remainder = 0 --> yes, x - 3 is a factor
7B. remainder = 1 -->no, x-3 is not a factor
8A. P(-2) = 77
8B. P(3) = 153
9A. (x + 1)(x - 1)(x - 3)
9B. (x - 3)(x - 3)(x + 2)
10A. for periods 5, 7:
max = 3.08 (at x = -.15) AND min = -3.08 (at x = 2.15)
10A. for period 6:
The polynomial from 9A doesn't really have a relative max/min.
Instead, try x^3 - 3x^2 - x + 3 and check the answers for periods 3, 5
10B. for periods 5, 7:
max = 0 (at x =0) AND min = -1.26 (at x = .82) AND min = -45.1 (at x = -3.07)
10B. for period 6:
max = 18.5 (at x = -.33) AND min = 0 (at x =3)
Or try x^4 + 3x^3 - 5x^2 and check the answers for periods 3, 5
11A. factored: 2(x + 3)(x^2 - 3x + 9) = 0
solutions: x = -3 AND x = (3 +/- 3i sqrt3)/2
11B. factored: (2x - 5)(4x^2 + 10x + 25) = 0
solutions: x = 5/2 AND x = (-5 +/- 5i sqrt3)/4
12A. factored: (x + 10)(x - 10) = 0
solutions: x = -10 AND x = 10
12B. factored: 3(x + 4)(x - 4) = 0
solutions: x = -4 AND x = 4
13A. x = +/- sqrt14 AND x = +/- i
13B. x = +/- i sqrt3 AND x = i sqrt5
14A. 3 - 2i and 1 - sqrt2
14B. 2 + sqrt3
15A. y = x^3 - 2x^2 + x - 2
15B. y = x^4 -10x^3 + 39x^2 - 70x + 50
16A. complex: 3, real: 1 or 3
possible rational roots: +/- 1, +/-3, +/-5, +/-15
16B. complex: 4, real: 0, 2, or 4
possible rational roots: +/-1, +/-2, +/-4, +/-1/3, +/-2/3, +/-4/3
17A. x = 0, x = +/-3, x = +/- 2i
17B. x = +/-2 sqrt2, x = +/- i sqrt6
18A. x = 4, x = +/- i sqrt7
18B. x = -2, x = 1 +/- sqrt7
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8 comments:
Mrs. Williams,
I'm confused about #10B. When I entered the 9B's equation into my graphing calculator, I got:
Max: 18.52 @ x=-.33
Min: 0 @ x=3
I don't understand how you got the answers posted on the blog. Help!
Thanks,
Wyatt Chan
y8chan@comcast.net
Mrs. Williams,
On number 9B, i have remainder of 18 after using the synthetic division. Also, on number 10B, my relative max and min are the same as wyatt's. Can you please show me number 9B and 10B please? I appreciate your time.
Thanks,
Vu Le
vule1607@yahoo.com
Regarding #9B:
Here's what my synthetic division looks like (the dots are just for spacing):
3 ...1 -4 -3 18
........3 -3 -18
.....1 -1 -6 0
This leaves x^2 - 1x - 6 = 0, which factors into (x - 3)(x + 2). So the original polynomial has three factors: (x - 3)(x - 3)(x + 2)
Regarding #10B:
My first guess is that maybe we are using different equations--maybe I wrote it on the board differently than I wrote it on MY paper. I am using the polynomial
x^4 + 3x^3 - 5x^2
I checked it again, and the relative max and mins are correct for this polynomial. Please let me know if I told you to use something different so that I can update the answers to reflect it!
Mrs. Williams,
I've found my mistake on #9b. Thank you. About #10B, I assumed that we have to use the equations from #9B, which is x^3-4x^2-3X+18, because there is nothing on #10B. Also, that polynomial x^4 + 3x^3 - 5x^2, I don't see it anywhere on the paper. Did you write it on the board and i forgot to copy it down?
Thanks
Vu Le
Sorry about the confusion on #10A and #10B! I forgot to include polynomials on those questions, and because 6th period was so short on Thursday, no one got there to point it out to me!
I have corrected the answers to reflect this mix-up.
Mrs. Williams,
On 9A and B, I am not quite sure what the question is asking me to do.
-Maggie
Regarding #9A and #9B:
For each polynomial, a known linear factor is given. Use synthetic division to divide the polynomial by the given factor, then factor the remaining quadratic.
The three linear factors (the given one, and the two you get from factoring the quadratic) should "completely factor" the original polynomial.
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