We are told that x³ - y³ = 56. We can factor the left side of the equation using the formula for difference of cubes.

x³ - y³ = (x - y)(x² + xy + y²) = 56

Since x - y = 2, we can substitute this value in for the factor x - y.

2(x² + xy + y²) = 56

Divide both sides by 2.

x² + xy + y² = 28

Because we are trying to find x² + y², if we can get rid of xy, then we would have our answer. 

We are told that 3xy = 24. If we divide both sides by 3, we see that xy = 8.

We can then substitute this value into the equation x² + xy + y² = 28.

x² + 8 + y² = 28

Subtract both sides by eight.

x² + y² = 20.

The answer is 20. 

ALTERNATE SOLUTION:

We are told that x - y = 2 and 3xy = 24. This is a system of equations. 

If we solve the first equation in terms of x, we can then substitute this into the second equation.

x - y = 2

Add y to both sides.

x = y + 2

Now we will substitute this value for x into the second equation.

3(y+2)(y) = 24

Now we can divide both sides by three.

(y+2)(y) = 8

Then we distribute.

y² + 2y = 8

Subtract 8 from both sides.

y² + 2y - 8 = 0

We need to factor this by thinking of two numbers that multiply to give -8 but add to give 2. These numbers are 4 and -2.

(y + 4)(y - 2) = 0

This means either y - 4 = 0, or y + 2 = 0

y = -4, or y = 2

Because x = y + 2, if y = -4, then x must be -2. Similarly, if y = 2, then x must be 4. 

Let's see which combination of x and y will satisfy the final equation that we haven't used, x³ - y³ = 56.

If x = -2 and y = -4, then

(-2)³ - (-4)³ = -8 - (-64) = 56. So that means that x= -2 and y = -4 is a valid solution.

If x = 4 and y = 2, then

(4)³ - 2³ = 64 - 8 = 56. So that means x = 4 and y = 2 is also a valid solution.

The final value we are asked to find is x² + y².

If x= -2 and y = -4, then x² + y² = (-2)² + (-4)² = 4 + 16 = 20.

If x = 4 and y = 2, then  x² + y² = (4)² + 2² = 16 + 4 = 20.

Thus, no matter which solution we use for x and y, x² + y² = 20.

The answer is 20. 

First, subtract 3 from both sides in order to obtain an equation that equals 0:

The left side can be factored. We need factors of  that add up to .  and  work:

Set both factors equal to 0 and solve:

To solve the left equation, add 1 to both sides. To solve the right equation, subtract 3 from both sides. This yields two solutions:

Only one of these solutions is negative, so the answer is 1.

 can be seen to fit the pattern 

:

where 

 can be factored as , so 

, as the sum of squares, is a prime polynomial, so the complete factorization is

,

making  the only prime factor, and "one" the correct choice.

To begin, factor out any like terms from the expression. In this case, the term  can be pulled out:

Next, recall the difference of squares:

Here, and .

Thus, our answer is

.

Add the two equations together to yield 5x + 5y = 9a + b, then factor out 5 to get 5(x + y) = 9a + b; divide both sides by 5 to get x + y = (9a + b)/5; subtract the two equations to get x - y = -a - 3b. So, x² – y² = (x + y)(x – y) = (9a + b)/5 (–a – 3b) = (–[(9a)]² – 28ab – [(3b)]²)/5

The polynomial has one term, so its degree is the sum of the exponents of the variables:

The degree of a polynomial in more than one variable is the greatest degree of any of the terms; the degree of a term is the sum of the exponents. The degrees of the terms in the given polynomial are:

The degree of the polynomial is the greatest of these degrees, 100.

The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 44, 20, 10, and 100; the greatest of these is 100, which is the degree.

The degree of a polynomial in one variable is the greatest exponent of any of the powers of the variable. The terms have as their exponents, in order, 10, 20, 30, and 40; 40 is the greatest of them and is the degree of the polynomial.

The degree of a polynomial is the highest degree of any term; the degree of a term is the exponent of its variable or the sum of the exponents of its variables, with unwritten exponents being equal to 1. For each term in a polynomial, write the exponent or add the exponents; the greatest number is its degree. We do this with all four choices:

:

 A constant term has degree 0.

The degree of this polynomial is 5.

The degree of this polynomial is 5.

The degree of this polynomial is 5.

The degree of this polynomial is 5.

All four polynomials have the same degree.