poly. fcn. with zeroes -3, -4+i, 1-sqrt[3]; factor 8c^3-512;

[PineappleExpress]

-Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros.
-3, -4 + i, 1-?3

Factor Polynomial completely
8c^3 - 512

Evaluate Expression
(1.5 x 10^4)(3.2 x 10^-8) <power of negative 8
(1.2 x 10^-7) <power of negative 7

Anyone help me out please :?

 

[soroban]

Re: Help for Homework

Hello, PineappleExpress!

Here's the first one . . .

Write a polynomial function \(\displaystyle f\) of least degree that has rational coefficients,
a leading coefficient of 1, and the given zeros: .\(\displaystyle -3,\;\;-4 + i,\;\;1-\sqrt{3}\)

Complex roots occur in conjugate pairs, so both \(\displaystyle -4 + i\text{ and }-4-i\) are roots.

Irrational roots occur in conjugate pairs, so both \(\displaystyle 1 + \sqrt{3}\text{ and }1 - \sqrt{3}\) are roots.


The least-degree polynomial is:

. . \(\displaystyle f(x) \;=\;\underbrace{\bigg[x - (-3)\bigg]}\,\underbrace{\bigg[x - (-4+i)\bigg]\,\bigg[x - (-4-i)\bigg]}\,\underbrace{\bigg[x - (1 + \sqrt{3})\bigg]\,\bigg[x - (1 - \sqrt{3})\bigg]}\)

. . . . . .\(\displaystyle = \quad\underbrace{\bigg[x + 3\bigg]\qquad\qquad\quad\bigg[x^2 + 8x + 17\bigg]\qquad\qquad\qquad \quad \bigg[x^2 - 2x - 2\bigg]}\)

. . . . . .\(\displaystyle = \qquad\qquad\qquad\quad x^5 + 9x^4 + 17x^2 - 53x^2 - 184x - 102\)

But check my algebra . . . please!


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


Here's a tip for this type of multiplication . . .


\(\displaystyle \text{Suppose we want: }\;\bigg[x - (3 + i)\bigg]\,\bigg[x - (3 - i)\bigg]\)

\(\displaystyle \text{Rewrite it like this: }\;\bigg[(x - 3)\; -\; i\bigg]\,\bigg[(x-3) \;+\; i\bigg]\quad\hdots\quad \text{It's of the form: }(a - b)(a + b)\)

\(\displaystyle \text{So we have: }\;(x-3)^2 - i^2 \;\;=\;\;x^2 - 6x + 9 + 1 \;\;=\;\;x^2 - 6x + 10\)

 

[PineappleExpress]

Re: Help for Homework

Thank you so much!
-I think you can combine the 17x^2 - 53x^2 but this helped me out so much!
-My test is tomorrow hopefully i can get an explanation on the other two soon.
-Thanks again for putting the time to write all this out i appreciate it a lot. :mrgreen:

 

[mmm4444bot]

… I think you can combine the 17x^2 - 53x^2 …

You think wrongly.

This is what happens when you rely on other people to do your thinking for you. Somebody makes an obvious typographical error; it blares out at you with a flashing red light, but you see it not (because you've never had to think much about these things).

I suggest that you follow soroban's instruction to check the algebra.

Multiply out the last factored form, yourself. Then you will know why you thought wrongly.

Are you familiar with the special factoring forms (eg: difference of squares, difference of cubes, sum of cubes)? The second exercise is one of these.

You typed "evaluate 1.2 x 10^-7" for part of the third exercise. There is nothing to evaluate.

1.2 x 10^-7 is scientific notation for the number 0.00000012, so you either dropped part of the exercise or you missed the instruction.

To multiply two numbers written in scientific notation, use the Commutative Property of Multiplication and rearrange the order.

EG

(2.6 x 10^4)(3.7 x 10^-5) = (2.6)(3.7) x (10^4)(10^-5) = 9.62 x 10^-1

Do you know how to multiply two powers together?

How many hours before your test did you realize that you do not understand much of this?

 

[PineappleExpress]

Re: Help for Homework

AH, thanks for correcting me on the first part =P
-Im sorry for the confusion for the evaluate equation its actually

(1.5 x 10^4)(3.2 x 10^-8) / (1.2 x 10^-7) <-- as in divided.

 

[Deleted member 4993]

Re: Help for Homework

AH, thanks for correcting me on the first part =P
-Im sorry for the confusion for the evaluate equation its actually

(1.5 x 10^4)(3.2 x 10^-8) / (1.2 x 10^-7) <-- as in divided.

hint:

\(\displaystyle \frac{a^m \cdot a^n}{a^p} \, = \, a^{m+n-p}\)

 

[Deleted member 4993]

Factor Polynomial completely
8c^3 - 512
Hint:

use
\(\displaystyle a^3 - b^3 \, = (a-b)\cdot (a^2+ab+b^2)\)

 

[fasteddie65]

Factor Polynomial completely
8c^3 - 512 = 8(c^3 - 64) This is a difference of cubes.

Evaluate Expression
(1.5 x 10^4)(3.2 x 10^-8) <power of negative 8 = 4.8 x 10^(-4)
(1.2 x 10^-7) <power of negative 7

4.8 x 10^(-4) / (1.2 x 10^-7 = 4 x 10^3 = 4000