# Tutor profile: David C.

## Questions

### Subject: Pre-Calculus

Use the Fundamental Theorem of Algebra to find how many roots are in \[f(x)=x^3-1\] and find the roots.

The highest power on the variable, x, is 3, so there are a total of 3 roots, though some may not be real roots. Find the roots by factoring \[x^3-1=0\] This is the difference of two cubes formula, \[a^3-b^3= (a-b)(a^2+ab+b^2)\] where a=x and b=1. Substituting yields \[x^3-1=(x-1)(x^2+x+1)\] So, solve \[(x-1)(x^2+x+1)=0\] The first factor gives \[x-1=0\], or x=1 The second is not factorable over the Real numbers, so use the quadratic formula. For ax^2+bx+c=0 x=(-b\pm \sqrt{}(b^2-4ac)/2a Substituting a=1, b=1, c=1 x=(-1\pm \sqrt{}(1^2-4(1)(1))/2(1) Thus, the two complex roots are (-1+\sqrt{3}i)/2 and (-1-\sqrt{3}i)/2

### Subject: Pre-Algebra

There are 8 classrooms of students in a school, and each class has the same number of students. The Principal assigns a smartphone to every student. Write an expression for the number of smartphones distributed.

Lets use "x" as the number of students in each class, since it will be the same in every class. To get the number of smartphones in total, we multiply 8 by the number of students in each class. If the number of students per class were 10, then the number of smartphones in total would be 8 (10). If the number of students per class were instead 12, the number of smartphones would be 8 (12). But, the number of students per class is "x", so the number of smartphones in total is 8x.

### Subject: Algebra

Travis bought 2 gallons of raw milk and 3 pounds of grassed butter, totaling $48.00. If the butter costs $8 per pound, how much does the milk cost?

If we let m=the number of gallons of milk, then the cost of all the milk would be the number of gallons bought times the price per gallon, or 2m. If we let b=number of pounds of butter purchased, then the cost of all the butter would be the number of pounds purchased times the cost per pound, or 3b. To find the total cost of the entire purchase, add the total cost of all the milk to the total cost of all the butter, or 2m + 3b. We know this will be $48.00. Therefore, we can make an equation for this problem: 2m + 3b = 48, where m is the cost of the milk per gallon and b is the cost of the butter per pound. Since we know the price of the butter is $8 per pound, substitute the 8 for the "b", giving 2m + 3(8) = 48. Simplifying yields 2m + 24 = 48, which is a one variable equation we can easily solve. Subtract 24 from both sides, yielding 2m + 24 - 24 = 48 - 24. Simplifying both sides, we get 2m = 24. Dividing both sides by 2 yields 2m/2 = 24/2, simplifying to m = 12. So, the cost of the milk per gallon is $12.

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