Algebra

To square the expression (3A + 8B), we can use the formula (x + y)² = x² + 2xy + y².

In this case, x = 3A and y = 8B.

So, (3A + 8B)² = (3A)² + 2(3A)(8B) + (8B)²

Now, we simplify each term:

• (3A)² = 9A²
• 2(3A)(8B) = 48AB
• (8B)² = 64B²

Therefore, (3A + 8B)² = 9A² + 48AB + 64B²

To solve the expression 1/(X+Y) + 2X/(X^2+Y^2) + 4X^3Y^4/(Y^8-X^8), we can simplify it step by step.

1. Rewrite the expression:

The given expression is:

1/(X+Y) + 2X/(X^2+Y^2) + 4X^3Y^4/(Y^8-X^8)

2. Factor the denominator of the third term:

Notice that (Y^8 - X^8) can be factored as a difference of squares:

Y^8 - X^8 = (Y^4 + X^4)(Y^4 - X^4) = (Y^4 + X^4)(Y^2 + X^2)(Y^2 - X^2) = (Y^4 + X^4)(Y^2 + X^2)(Y + X)(Y - X)

3. Rewrite the expression using the factored form:

1/(X+Y) + 2X/(X^2+Y^2) + 4X^3Y^4/((Y^4+X^4)(Y^2+X^2)(Y+X)(Y-X))

4. Combine the first two terms:

Find a common denominator for the first two terms, which is (X+Y)(X^2+Y^2):

[1/(X+Y)] + [2X/(X^2+Y^2)] = [(X^2+Y^2) + 2X(X+Y)] / [(X+Y)(X^2+Y^2)]

= (X^2 + Y^2 + 2X^2 + 2XY) / ((X+Y)(X^2+Y^2))

= (3X^2 + 2XY + Y^2) / ((X+Y)(X^2+Y^2))

5. Look for a pattern or further simplification:

Now we have:

(3X^2 + 2XY + Y^2) / ((X+Y)(X^2+Y^2)) + 4X^3Y^4/((Y^4+X^4)(Y^2+X^2)(Y+X)(Y-X))

This expression is quite complex and does not appear to simplify easily into a compact form.

6. Alternative approach

Consider combining the first and second terms like so:

1/(X+Y) + 2X/(X^2+Y^2) = (X^2+Y^2+2X(X+Y))/((X+Y)(X^2+Y^2))

= (X^2+Y^2+2X^2+2XY)/((X+Y)(X^2+Y^2))

= (3X^2+2XY+Y^2)/((X+Y)(X^2+Y^2))

Which doesn't lead to a clear simplification when combined with the third term.

Given the complexity, it's likely that the problem is designed to be simplified using a specific substitution or context that isn't provided. If there's no further context, the expression remains as is or might require numerical methods for specific X and Y values.

A polynomial is a mathematical expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Here's a breakdown of the components:

• Variables: These are symbols (usually letters like x, y, z) that represent unknown or changeable values.
• Coefficients: These are the numbers that multiply the variables. They can be any real number (e.g., 2, -5, 1/2, √3).
• Exponents: These are the powers to which the variables are raised. In a polynomial, exponents must be non-negative integers (e.g., 0, 1, 2, 3, ...).

Examples of Polynomials:

• 3x² + 2x - 1
• y⁴ - 7y + 6
• 5z
• 8 (This is a constant polynomial, where the variable has an exponent of 0)

Examples of Non-Polynomials:

• 3x^-2 (Negative exponent)
• 2√x (Fractional exponent, as √x = x^1/2)
• 1/x (Variable in the denominator, which is equivalent to x^-1)
• sin(x) (Trigonometric function)

Polynomials can have one or more variables. A polynomial with one variable is called a "polynomial in one variable."

Source: MathWorld - Polynomial

Given that x = 6 and y = 3, we need to find the value of the expression:

(2xy + 7y - 10) / (4xy - 3x - 2)

First, let's substitute the values of x and y into the expression:

Numerator: 2xy + 7y - 10 = 2 * 6 * 3 + 7 * 3 - 10

2 * 6 * 3 = 36

7 * 3 = 21

So, the numerator becomes: 36 + 21 - 10 = 57 - 10 = 47

Denominator: 4xy - 3x - 2 = 4 * 6 * 3 - 3 * 6 - 2

4 * 6 * 3 = 72

3 * 6 = 18

So, the denominator becomes: 72 - 18 - 2 = 54 - 2 = 52

Therefore, the expression becomes: 47 / 52

So, the value of the expression is 47/52.