I had a mathematics test yesterday, and got the results back today. The quiz was on radicals, algebraic functions, and related. One question I got incorrect is now stumbling me.
Below is the question, then below that is the work I put in, the underlined step is the result I had:
Simply the following expressions.
f) 4x(x-2)^2
=> 4x(x-2)^2
=> 4x(x^2-4)
=> 4x^3-16x
However, the instructor had marked it incorrect, and put in red the supposed correct answer:
4x^3 - 16x^2 + 16x
How the fuck is that the answer? I tried complaining to the instructor, and even gave proof:
If I substitute the variable (x) for 1, and solve for the question, and answer she gave; they do not equal.
f) 4x(x-2)^2
=> 4(1-2)^2
=> 4
-. -) 4x^3 - 16x^2 + 16x
=> 4^3 - 16^2 + 16
=> 64 - 256 + 16
=> -176
Since 4 != -176, the answer she gave cannot be correct. Am I making a mistake, or am I right?
First off, in your “proof,” you completely misuse exponents. If you substitued 1 for x, it would be 4(1)^3 - 16(1)^2 + 16(1).
Now, we all (should) know 1 raised to any power is 1. So, we can simplify that to
4 - 16 + 16, which does indeed equal 4.
Secondly, her answer is right. Proof!
4x(x-2)^2
=> 4x(x-2)(x-2)
=>4x(x^2 - 4x + 4)
=> 4x^3 - 16x^2 + 16x
Your problem is you tried distributing the original exponent into the parantheses. You can’t just slide it in. You have to treat the entire object as being squared, so you have to multiply the object (x-2) by itself, which gives you x^2 - 4x + 4
It’s important to note that (a+b)^2 = (a+b)(a+b), a form you may be more familiar with.
Equations of the form (a+b)(c+d) must be multiplied in a special way. Traditionally students are taught to use the “FOIL” method or “CAT” method to do so, either of which you might recall from previous lectures. Using either method to multiply will give the result (a+b)(c+d) = ac+ad+bc+bd.
Because in this instance you have (a+b)(a+b), multiplying will give the result aa +ab+ba+bb which can be simplified as a^2+2ab+b^2.
The mnemonic device taught in my schools was BFEDMAS. Which doesn’t make a phrase, but is Brackets, factorials, exponents, division, multiplication, addition, and subtraction.
BFEDMAS? That’s silly. Parentheses are used for the same reason as brackets, but parentheses are more common. And how much do you run into factorials? And I think most people intuitively realize that a unary operator (like the factorial symbol) has higher precedence than a binary operator.
It’s just plain BEDMAS. I don’t think you do factorials much in highschool math unless you’re taking Finite. Even then it’s only in combinations and permutations.
The Wikipedia article for Order of Operations does mention factorials, Lex. PFEMDAS would be the correct order if we included F in the device. However, it’s such an obscure function for most people that we don’t include it. So, the textbook is just mentioning an obscure version of the one we all know and love.
If you press Ctrl and F on that Wikipedia page, and search for BFEDMAS, or PFEMDAS, it’s in the article. And the google link you gave directly points to that section in Wikipedia with BFEDMAS.
