Algebra Explained
Simplifying Polynomials-FOIL Method
FOIL stands for First, Outer, Inner, Last, in the order that they are conducted in, applying to simplifying expressions with two binomials each. What this means is that you multiply the first term in each binomial, then the outer terms, followed by the inner terms, and, as you may have guessed, the last terms.
>For example, (2x+4)(x-4) simplified is 2x^2-4x-16, with x^2 referring to x squared. We get this answer by doing as follows:
>Multiply 2x by x (the first term in each binomial).
>By multiplying (one) x with another (one) x, we will always recieve x^2. The 2 itself does not change. If one of the variables had an exponent, such as 2x^3 multiplied by x, the result would be 2x^4.
Multiply 2x by -4.
This is a much more simple equation than the former. 2x multiplied by -4 is simply -8x. Because the next step is to add 4, we end up with -4x. So we have 2x^2-4x. But how do we get -16?
Multiply 4 (-4).
4 times -4 is -16, and now we're done!
But how do we solve more complicated expressions?
For example, an expression with exponents before simplifying can be scary to look at, such as (x^2-8)(x-3). But fear not, for these are just as fun to simplify!
As usual, we start by identifying the terms. x^2 and x are our first terms, x^2 and 3 are our outer terms, 8 and x are the inner terms, and 8 and 3 are last. Right there, we just figured out what we need to do!
Multiplying x^2 by x.
As I have previously stated, multiplying variables (like x) is incredibly easy. It's essentially just addition, but instead of adding a coefficient, you add an exponent. So x with an exponent of 4 multiplied by x^3 is x^7. Simply add the exponents. All numbers, including variables, have an exponent of one, even if it's not said, which is why multiplying x and another x equals x^2. In this case, x^2 times x is x^3.
Multiplying x^2 by -3.
You might be able to guess this on your own, but the answer is just 3x^2. We can't multiply 3 by a variable, so we just attach it to the 3 for the time being.
-8 times x.
Like the previous step, we cannot multiply 8 by a variable, so we just attach it. So far, we have x^3-3x^2-8x.
Multiplying -8 by -3.
This last step is once again very easy, 8 times 3 is 24, and because both numbers are negative, the product is positive. The final result is x^3-3x^2-8x+24.
Calculus Explained
The most important destinction (in my opinion) between calculus and algebra is that rather than observing equations and expressions for solutions (as one does in algebra), the goal of most calculus problems is to observe the change. Noticing this, combined with an understanding of the symbols shown in most problems, might make calculus easier.
Because calculus is its own subject, there's many different types of problems and just saying 'calculus explained' doesn't really narrow it down, but I'll try to cover most basic problems you might encounter.
Let's try a basic problem, f(x)-x^2-8x-9, and feel free to follow this step-by-step video, courtesy of TabletClass Math :)
We'll be finding the first derivative in this function today, just to keep it simple. Generally, the symbol f'(x) refers to the first derivative, and f''(x) the second derivative. We'll start by splitting this into terms. -x^2, -8x, and -9.
Multiply the exponent by x's coefficient.
You'll notice that although x has an exponent of 2, it has no coefficient that we can see. This gives it a coefficient of 1. Because x is negative, so is the 1. 2 multiplied by itself once gives us just -2. Whatever the exponent is, we decrease it by one at this time. If it was 3, we would have an exponent of 2. Because it's just 2, we have 1, which we don't need to bother writing. We keep the x, and now we have our first term, -2x!
Multiply -8x.
-8x here has a power of 1, and -8 multiplied by itself once is still just -8. Drop the power on x one, and that's x to the zero, so we can get rid of the x, which gives us our second and final term, -8 :)
But JJ, I hear you cry, what happened to the -9? We got rid of it, just like the x on -8, because it had a power of 0.
So now we have our slope. But what if we need exact coordinates of the function's vertex? We can find that too!
We'll rewrite the slope we found as -2x-8=0. We'll call the vertex 0, because the tippy-top (or the bottom, if the parabola opens upward) is where the slope, or change, equals zero. Move the -8 to the other side of the equation, -2x=8. Divide both sides by -2, which gives us -4. So x=-4.
To find the exact coordinate, including y, we'll put -4 into the original function, like so: f(-4)=-(-4)^2-8(-4)-9, and solve it piece by piece. -(-4)^2 is -16, -8(-4) is a positive 32, so we now have -16+32-9, and by smushing all those together, we recieve 7. So the coordinates of this function is x=-4, y=7 :)
And you've just solved a calculus problem! Hooray for you!
