The boy would have 3 brothers and 3 sisters therefore in the family there would be 4 boys and 3 girls
PERT = ![\frac{OT+4MLT+PT}{6}](https://tex.z-dn.net/?f=%5Cfrac%7BOT%2B4MLT%2BPT%7D%7B6%7D)
Activity B: OT = 3, MLT = 6, PT = 12
PERT = ![\frac{3+4(6)+12}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B3%2B4%286%29%2B12%7D%7B6%7D)
= ![\frac{3+24+12}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B3%2B24%2B12%7D%7B6%7D)
= ![\frac{39}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B39%7D%7B6%7D)
= 6.5
Answer:
90
Step-by-step explanation:
The most appropriate choice for polynomial will be given by
1) Zeroes of P(x) = 2,
, ![\frac{2 - \sqrt{2}i}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B2%20-%20%5Csqrt%7B2%7Di%7D%7B3%7D)
where ![i = \sqrt{-1}](https://tex.z-dn.net/?f=i%20%3D%20%5Csqrt%7B-1%7D)
2) Zeroes of P(x) = 3, 2i, -2i
3) Roots are 2i, -2i, ![\frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B2%7D)
4) Roots are 0, 1,
, ![2 - \sqrt{5}](https://tex.z-dn.net/?f=2%20-%20%5Csqrt%7B5%7D)
What is a polynomial?
An algebraic expression of the form
is called a polynomial of degree n.
![1) P(x ) = 3x^3 -10x^2 + 10x -4\\P(2) = 3(2)^3 - 10(2)^2 +10(2) - 4\\](https://tex.z-dn.net/?f=1%29%20P%28x%20%29%20%3D%203x%5E3%20-10x%5E2%20%2B%2010x%20-4%5C%5CP%282%29%20%3D%203%282%29%5E3%20-%2010%282%29%5E2%20%2B10%282%29%20-%204%5C%5C)
![= 24 -40 + 20 -16\\= 0](https://tex.z-dn.net/?f=%3D%2024%20-40%20%2B%2020%20-16%5C%5C%3D%200)
(x - 2) is a factor of P(x)
![P(x) = 3x^2(x - 2) -4x(x - 2) +2(x-2)\\](https://tex.z-dn.net/?f=P%28x%29%20%3D%203x%5E2%28x%20-%202%29%20-4x%28x%20-%202%29%20%2B2%28x-2%29%5C%5C)
= ![(x - 2)(3x^2 - 4x + 2)](https://tex.z-dn.net/?f=%28x%20-%202%29%283x%5E2%20-%204x%20%2B%202%29)
![=(x-2)(x -a)(x - b)](https://tex.z-dn.net/?f=%3D%28x-2%29%28x%20-a%29%28x%20-%20b%29)
where,
![a = \frac{-(-4)+\sqrt{(-4)^2 - 4\times 3\times 2}}{2\times 3}\\a =\frac{ 4 + \sqrt{-8}}{6}\\a = \frac{4 + 2\sqrt{2} i}{6}\\a = \frac{2(2 + \sqrt{2}i)}{6}\\a = \frac{2 + \sqrt{2}i}{3}](https://tex.z-dn.net/?f=a%20%3D%20%5Cfrac%7B-%28-4%29%2B%5Csqrt%7B%28-4%29%5E2%20-%204%5Ctimes%203%5Ctimes%202%7D%7D%7B2%5Ctimes%203%7D%5C%5Ca%20%3D%5Cfrac%7B%204%20%2B%20%5Csqrt%7B-8%7D%7D%7B6%7D%5C%5Ca%20%3D%20%5Cfrac%7B4%20%2B%202%5Csqrt%7B2%7D%20i%7D%7B6%7D%5C%5Ca%20%3D%20%5Cfrac%7B2%282%20%2B%20%5Csqrt%7B2%7Di%29%7D%7B6%7D%5C%5Ca%20%3D%20%5Cfrac%7B2%20%2B%20%5Csqrt%7B2%7Di%7D%7B3%7D)
![b = \frac{-(-4)-\sqrt{(-4)^2 - 4\times 3\times 2}}{2\times 3}\\b =\frac{ 4 -\sqrt{-8}}{6}\\b = \frac{4 - 2\sqrt{2} i}{6}\\b = \frac{2(2 - \sqrt{2}i)}{6}\\b = \frac{2 - \sqrt{2}i}{3}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B-%28-4%29-%5Csqrt%7B%28-4%29%5E2%20-%204%5Ctimes%203%5Ctimes%202%7D%7D%7B2%5Ctimes%203%7D%5C%5Cb%20%3D%5Cfrac%7B%204%20-%5Csqrt%7B-8%7D%7D%7B6%7D%5C%5Cb%20%3D%20%5Cfrac%7B4%20-%202%5Csqrt%7B2%7D%20i%7D%7B6%7D%5C%5Cb%20%3D%20%5Cfrac%7B2%282%20-%20%5Csqrt%7B2%7Di%29%7D%7B6%7D%5C%5Cb%20%3D%20%5Cfrac%7B2%20-%20%5Csqrt%7B2%7Di%7D%7B3%7D)
Zeroes of P(x) = 2,
, ![\frac{2 - \sqrt{2}i}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B2%20-%20%5Csqrt%7B2%7Di%7D%7B3%7D)
where ![i = \sqrt{-1}](https://tex.z-dn.net/?f=i%20%3D%20%5Csqrt%7B-1%7D)
![2) P(x) = x^3 - 3x^2+4x-12\\P(3) = (3)^3 - 3(3)^2 +4(3) -12\\ P(3) = 0](https://tex.z-dn.net/?f=2%29%20P%28x%29%20%3D%20x%5E3%20-%203x%5E2%2B4x-12%5C%5CP%283%29%20%3D%20%283%29%5E3%20-%203%283%29%5E2%20%2B4%283%29%20-12%5C%5C%20%20%20%20%20P%283%29%20%3D%200)
(x - 3) is a factor of P(x)
![x^2(x - 3) + 4(x - 3)\\(x - 3)(x^2 + 4)\\(x - 3)(x -a)(x-b)\\](https://tex.z-dn.net/?f=x%5E2%28x%20-%203%29%20%2B%204%28x%20-%203%29%5C%5C%28x%20-%203%29%28x%5E2%20%2B%204%29%5C%5C%28x%20-%203%29%28x%20-a%29%28x-b%29%5C%5C)
where,
![a = \sqrt{-4}\\a = 2i](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%7B-4%7D%5C%5Ca%20%3D%202i)
![b = -\sqrt{-4}\\a = -2i](https://tex.z-dn.net/?f=b%20%3D%20-%5Csqrt%7B-4%7D%5C%5Ca%20%3D%20-2i)
Zeroes of P(x) = 3, 2i, -2i
![3) 2x^3 - 3x^2 +8x-12= 0\\](https://tex.z-dn.net/?f=3%29%202x%5E3%20-%203x%5E2%20%2B8x-12%3D%200%5C%5C)
x = 2 satisfies the equation
![2x^2(x -\frac{3}{2}) + 8(x-\frac{3}{2})=0\\(2x^2+8)(x - \frac{3}{2}) = 0\\](https://tex.z-dn.net/?f=2x%5E2%28x%20-%5Cfrac%7B3%7D%7B2%7D%29%20%2B%208%28x-%5Cfrac%7B3%7D%7B2%7D%29%3D0%5C%5C%282x%5E2%2B8%29%28x%20-%20%5Cfrac%7B3%7D%7B2%7D%29%20%3D%200%5C%5C)
or ![x - \frac{3}{2} = 0](https://tex.z-dn.net/?f=x%20-%20%5Cfrac%7B3%7D%7B2%7D%20%3D%200)
or ![x = \frac{3}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B3%7D%7B2%7D)
or ![x = \frac{3}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B3%7D%7B2%7D)
or ![x = \frac{3}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B3%7D%7B2%7D)
or
or ![x = \frac{3}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B3%7D%7B2%7D)
Roots are 2i, -2i, ![\frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B2%7D)
4)
![x^4 - 5x^3 +3x^2 +x = 0\\x(x^3 -5x^2 + 3x +1) = 0\\](https://tex.z-dn.net/?f=x%5E4%20-%205x%5E3%20%20%2B3x%5E2%20%2Bx%20%3D%200%5C%5Cx%28x%5E3%20-5x%5E2%20%2B%203x%20%2B1%29%20%3D%200%5C%5C)
or ![x^3 -5x^2+3x +1 = 0](https://tex.z-dn.net/?f=x%5E3%20-5x%5E2%2B3x%20%2B1%20%3D%200)
For ![x^3 -5x^2+3x +1 = 0](https://tex.z-dn.net/?f=x%5E3%20-5x%5E2%2B3x%20%2B1%20%3D%200)
x = 1 satisfies the equation
![x^2(x -1) -4x(x-1)-1(x-1) = 0\\(x - 1)(x^2 - 4x -1) = 0\\](https://tex.z-dn.net/?f=x%5E2%28x%20-1%29%20-4x%28x-1%29-1%28x-1%29%20%3D%200%5C%5C%28x%20-%201%29%28x%5E2%20-%204x%20-1%29%20%3D%200%5C%5C)
or ![x^2 - 4x -1 = 0](https://tex.z-dn.net/?f=x%5E2%20-%204x%20-1%20%3D%200)
Roots are x = 1 or x = a or x = b
where,
![a = \frac{-(-4) + \sqrt{(-4)^2 - 4\times 1 \times(-1)}}{2\times 1}\\a = \frac{4+\sqrt{20}}{2}\\a = \frac{4 + 2\sqrt{5}}{2}\\a = \frac{2(2 + \sqrt{5})}{2}\\a = 2 + \sqrt{5}](https://tex.z-dn.net/?f=a%20%3D%20%5Cfrac%7B-%28-4%29%20%2B%20%5Csqrt%7B%28-4%29%5E2%20-%204%5Ctimes%201%20%5Ctimes%28-1%29%7D%7D%7B2%5Ctimes%201%7D%5C%5Ca%20%3D%20%5Cfrac%7B4%2B%5Csqrt%7B20%7D%7D%7B2%7D%5C%5Ca%20%3D%20%5Cfrac%7B4%20%2B%202%5Csqrt%7B5%7D%7D%7B2%7D%5C%5Ca%20%3D%20%5Cfrac%7B2%282%20%2B%20%5Csqrt%7B5%7D%29%7D%7B2%7D%5C%5Ca%20%3D%202%20%2B%20%5Csqrt%7B5%7D)
![b = \frac{-(-4) - \sqrt{(-4)^2 - 4\times 1 \times(-1)}}{2\times 1}\\b = \frac{4-\sqrt{20}}{2}\\b = \frac{4 - 2\sqrt{5}}{2}\\b = \frac{2(2 - \sqrt{5})}{2}\\b = 2 - \sqrt{5}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B-%28-4%29%20-%20%5Csqrt%7B%28-4%29%5E2%20-%204%5Ctimes%201%20%5Ctimes%28-1%29%7D%7D%7B2%5Ctimes%201%7D%5C%5Cb%20%3D%20%5Cfrac%7B4-%5Csqrt%7B20%7D%7D%7B2%7D%5C%5Cb%20%3D%20%5Cfrac%7B4%20-%202%5Csqrt%7B5%7D%7D%7B2%7D%5C%5Cb%20%3D%20%5Cfrac%7B2%282%20-%20%5Csqrt%7B5%7D%29%7D%7B2%7D%5C%5Cb%20%3D%202%20-%20%5Csqrt%7B5%7D)
Roots are 0, 1,
, ![2 - \sqrt{5}](https://tex.z-dn.net/?f=2%20-%20%5Csqrt%7B5%7D)
To learn more about polynomial, refer to the link-
brainly.com/question/2833285
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