Answer:
it demonstrates the power property
![{ log(a) }^{2} = 2 log(a)](https://tex.z-dn.net/?f=%20%20%7B%20log%28a%29%20%7D%5E%7B2%7D%20%20%3D%202%20log%28a%29%20)
4 x 1 thousand (1000) = 4000
21 x 1 ten (10) = 210
4000
+ 210
---------
4210
=4210 (four thousand-two hundred and ten
<h3>Zeros of function are
![x = 1 + \sqrt{2} \text{ and } x = 1 - \sqrt{2}](https://tex.z-dn.net/?f=x%20%3D%201%20%2B%20%5Csqrt%7B2%7D%20%5Ctext%7B%20and%20%7D%20x%20%3D%201%20-%20%5Csqrt%7B2%7D)
</h3>
<em><u>Solution:</u></em>
<em><u>We have to find the zeros of the function</u></em>
![y = -x^2 + 2x+1](https://tex.z-dn.net/?f=y%20%3D%20-x%5E2%20%2B%202x%2B1)
Find the zeros of function:
![-x^2 + 2x+1 = 0\\\\\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=-x%5E2%20%2B%202x%2B1%20%3D%200%5C%5C%5C%5C%5Cmathrm%7BFor%5C%3Aa%5C%3Aquadratic%5C%3Aequation%5C%3Aof%5C%3Athe%5C%3Aform%5C%3A%7Dax%5E2%2Bbx%2Bc%3D0%5Cmathrm%7B%5C%3Athe%5C%3Asolutions%5C%3Aare%5C%3A%7D%5C%5C%5C%5Cx%3D%5Cfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
![\mathrm{For\:}\quad a=-1,\:b=2,\:c=1\\\\x =\frac{-2\pm \sqrt{2^2-4\left(-1\right)1}}{2\left(-1\right)}](https://tex.z-dn.net/?f=%5Cmathrm%7BFor%5C%3A%7D%5Cquad%20a%3D-1%2C%5C%3Ab%3D2%2C%5C%3Ac%3D1%5C%5C%5C%5Cx%20%20%3D%5Cfrac%7B-2%5Cpm%20%5Csqrt%7B2%5E2-4%5Cleft%28-1%5Cright%291%7D%7D%7B2%5Cleft%28-1%5Cright%29%7D)
![Simplify\\\\x=\frac{-2 \pm \sqrt{4+4}}{-2}\\\\x =\frac{-2 \pm \sqrt{8}}{-2}\\\\Simplify\\\\x =\frac{-2 \pm 2 \sqrt{2}}{-2}\\\\x = 1 \pm \sqrt{2}](https://tex.z-dn.net/?f=Simplify%5C%5C%5C%5Cx%3D%5Cfrac%7B-2%20%5Cpm%20%5Csqrt%7B4%2B4%7D%7D%7B-2%7D%5C%5C%5C%5Cx%20%3D%5Cfrac%7B-2%20%5Cpm%20%5Csqrt%7B8%7D%7D%7B-2%7D%5C%5C%5C%5CSimplify%5C%5C%5C%5Cx%20%3D%5Cfrac%7B-2%20%5Cpm%202%20%5Csqrt%7B2%7D%7D%7B-2%7D%5C%5C%5C%5Cx%20%3D%201%20%5Cpm%20%5Csqrt%7B2%7D)
We have two zeros
![x = 1 + \sqrt{2} \text{ and } x = 1 - \sqrt{2}](https://tex.z-dn.net/?f=x%20%3D%201%20%2B%20%5Csqrt%7B2%7D%20%5Ctext%7B%20and%20%7D%20x%20%3D%201%20-%20%5Csqrt%7B2%7D)
Thus zeros of function are ![x = 1 + \sqrt{2} \text{ and } x = 1 - \sqrt{2}](https://tex.z-dn.net/?f=x%20%3D%201%20%2B%20%5Csqrt%7B2%7D%20%5Ctext%7B%20and%20%7D%20x%20%3D%201%20-%20%5Csqrt%7B2%7D)
The details of the packages are not posted, however I can tell you how to find the answer.
First determine how many pencils are in each pack.
Then divide the price of the pack by the number of pencils to find out which pack has the least cost per pencil. To put this in equation form:
![\frac{Cost}{Pencils}](https://tex.z-dn.net/?f=%20%5Cfrac%7BCost%7D%7BPencils%7D%20)
Whichever pack has the least cost per pencil is the one that is the best priced.
<u>Assuming that you want the product of the two factors given in each problem</u>:
⇒ we solve this problem by timing each unit of a factor to every unit
of the other factor
⇒then add each one of them who shares a common like-term
⇒to get the answer
<u>Let's solve:</u>
![(4m+n)(m-2n)=4m*m+(-2n)*4m+n*m+n(-2n)\\=4m^2-7nm-2n^2](https://tex.z-dn.net/?f=%284m%2Bn%29%28m-2n%29%3D4m%2Am%2B%28-2n%29%2A4m%2Bn%2Am%2Bn%28-2n%29%5C%5C%3D4m%5E2-7nm-2n%5E2)
![(2a-4b)(7a-2b)=2a*7a+(-2b)(2a)+(-4b)7a+(-4b)(-2b)\\=14a^2-4ab-28ab+8b^2\\=14a^2-32ab+8b^2](https://tex.z-dn.net/?f=%282a-4b%29%287a-2b%29%3D2a%2A7a%2B%28-2b%29%282a%29%2B%28-4b%297a%2B%28-4b%29%28-2b%29%5C%5C%3D14a%5E2-4ab-28ab%2B8b%5E2%5C%5C%3D14a%5E2-32ab%2B8b%5E2)
![(4m+1)^2=(4m+1)(4m+1)=4m*4m+4m*1+1*4m+1\\=16m^2+4m+4m+1\\=16m^2+8m+1](https://tex.z-dn.net/?f=%284m%2B1%29%5E2%3D%284m%2B1%29%284m%2B1%29%3D4m%2A4m%2B4m%2A1%2B1%2A4m%2B1%5C%5C%3D16m%5E2%2B4m%2B4m%2B1%5C%5C%3D16m%5E2%2B8m%2B1)
![(2w+1)(2w-1)=2w*2w+2w(-1)+1*(2w)+1(-1)\\=4w^2-1](https://tex.z-dn.net/?f=%282w%2B1%29%282w-1%29%3D2w%2A2w%2B2w%28-1%29%2B1%2A%282w%29%2B1%28-1%29%5C%5C%3D4w%5E2-1)
![(m+3)(m^2+4m+7)\\=m*m^2+m*4m+m*7+3m^2+3*4m+3*7\\=m^3+4m^2+7m+3m^2+12m+21\\=m^3+7m^2+19m+21](https://tex.z-dn.net/?f=%28m%2B3%29%28m%5E2%2B4m%2B7%29%5C%5C%3Dm%2Am%5E2%2Bm%2A4m%2Bm%2A7%2B3m%5E2%2B3%2A4m%2B3%2A7%5C%5C%3Dm%5E3%2B4m%5E2%2B7m%2B3m%5E2%2B12m%2B21%5C%5C%3Dm%5E3%2B7m%5E2%2B19m%2B21)
![(3x+2)(5x^2-12x-2)\\=3x*5x^2+3x(-12x)+3x(-2)+2(5x^2)+2(-12x)+2(-2)\\=15x^3-36x^2-6x+10x^2-24x-4\\=15x^3-26x^2-30x-4](https://tex.z-dn.net/?f=%283x%2B2%29%285x%5E2-12x-2%29%5C%5C%3D3x%2A5x%5E2%2B3x%28-12x%29%2B3x%28-2%29%2B2%285x%5E2%29%2B2%28-12x%29%2B2%28-2%29%5C%5C%3D15x%5E3-36x%5E2-6x%2B10x%5E2-24x-4%5C%5C%3D15x%5E3-26x%5E2-30x-4)
Hope that helps!