Let's try to factor x² – y² – z² + 2yz.

Notice that the last three terms are very close to y² + z² – 2yz, which, if we rearranged them, would become y² – 2yz+ z². We could factor y² – 2yz+ z² as (y – z)², using the general rule that p² – 2pq + q² = (p – q)² .

So we want to rearrange the last three terms. Let's group them together first.

x² + (–y² – z² + 2yz)

If we were to factor out a –1 from the last three terms, we would have the following:

x² – (y² + z² – 2yz)

Now we can replace y² + z² – 2yz with (y – z)².

x² – (y – z)²

This expression is actually a differences of squares. In general, we can factor p² – q² as (p – q)(p + q). In this case, we can substitute x for p and (y – z) for q.

x² – (y – z)² = (x – (y – z))(x  + (y – z))

Now, let's distribute the negative one in the trinomial x – (y – z)

(x – (y – z))(x  + (y – z)) 

(x – y + z)(x + y – z)

The problem said that factoring x² – y² – z² + 2yz would result in two polynomials in the form (ax + by + cz)(dx + ey + fz), where a, b, c, d, e, and f were all integers, and a > 0.

(x – y + z)(x + y – z) fits this form. This means that a = 1, b = –1, c = 1, d = 1, e = 1, and f = –1. The sum of all of these is 2.

The answer is 2. 

is a difference of squares.

The difference of squares formula is .

Therefore, = .

We can first factor out :

This factors further because there is a difference of squares:

You can solve this problem by substituting in  for  and solving for :

That means that  Fahrenheit is the same as  Celsius.

Group all the terms with the  variable.

Pull out an  term from parentheses.

There are no more common factors.

The correct answer is:  

This word problem can be broken down into a basic algebraic equation:

A half-loaded semi weighs 37,000 lbs. Subtracting the weight of the truck, we can determine that a half load weighs 17,000 lbs:

From that, we can determine that a full load = 34,000 lbs: 

Knowing the weight of a full load, we can calculate the weight of a 3/4 load:

Add the weight of the truck to get the total weight:

(300)(400) = 120,000 or 12 * 10⁴.

The three two multiply to become 8 and the powers of ten can be added to become 10²¹.

1. Solve for x in 3^x = 27. x = 3 because 3 * 3 * 3 = 27.
2. Since x = 3, one can substitute x for 3 in 2^2x 
3. Now, the expression is 2^2*3
4. This expression can be interpreted as 2^{2 *} 2² * 2². Since 2² = 4, the expression can be simplified to become 4 * 4 * 4 = 64.
5. You can also multiply the powers to simplify the expression. When you multiply the powers, you get 2⁶, or 2 * 2 * 2 * 2 * 2 * 2
6. 2⁶ = 64.

In order to solve this equation, we first need to find a common base for the exponents. We know that 2³ = 8 and 2⁴ = 16, so it makes sense to use 2 as a common base, and then rewrite each side of the equation as a power of 2.

8^x-3 = (2³)^x-3

We need to remember our property of exponents which says that (a^b)^c = a^bc.

Thus (2³)^x-3 = 2^3(x-3) = 2^{3x - 9}.

We can do the same thing with 16^4-x.

16^4-x = (2⁴)^4-x = 2^4(4-x) = 2^16-4x.

So now our equation becomes

2^{3x - 9} = 2^16-4x

In order to solve this equation, the exponents have to be equal. 

3x - 9 = 16 - 4x

Add 4x to both sides.

7x - 9 = 16

Add 9 to both sides.

7x = 25

Divide by 7.

x = 25/7.