91Ó°ÊÓ

The substitution method involves replacing variables in an expression or polynomial with specific values. It helps in simplifying expressions and evaluating them with given values.
In the original exercise, we are given the polynomial \(x^{2}y^{2} + 2xy - x\). To evaluate it when \(x = -4\) and \(y = 3\), substitute \(x\) with \(-4\) and \(y\) with \(3\).
- The substituted polynomial becomes \((-4)^2 (3)^2 + 2(-4)(3) - (-4)\),
- Each variable is replaced, transforming the polynomial into an arithmetic problem.
This approach simplifies complex polynomials into simpler sequences of arithmetic operations that are much easier to manage.

Exponentiation is the process of raising a number to the power of another number. It is a shorthand way to express repeated multiplication of the same factor. In mathematical terms, \(a^b\) means multiplying \(a\) by itself \(b\) times.
In our polynomial, glancing at \((-4)^2\) and \((3)^2\), exponentiation is necessary to simplify these terms.
- Calculating \((-4)^2\) gives \(16\),
- Similarly, \((3)^2\) results in \(9\).
Exponentiation helps in reducing parts of the polynomial to simpler numbers before proceeding with other operations.

The order of operations is crucial in mathematics to ensure expressions are evaluated consistently. It is commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
In the polynomial evaluation process, following the order of operations ensures that you simplify correctly:
- First, handle any parentheses, though they often just serve to clarify expressions rather than contain additional calculations.
- Next, perform exponentiation, as seen with \((-4)^2\) and \((3)^2\).
- Then, move on to multiplication, for example, \(16 \times 9\) and \(2 \times (-4) \times 3\).
- Finally, complete the calculation by adding or subtracting values in sequence, such as \(144 - 24 + 4\).
This methodological approach simplifies the process and avoids errors.

Arithmetic operations are basic math processes including addition, subtraction, multiplication, and division. They form the foundation of solving algebraic expressions after tasks like substitution and exponentiation have been completed.
In our polynomial, we perform several arithmetic operations:

• Multiply the results of the exponents: \(16 \times 9 = 144\).
• Perform the multiplication: \(2 \times (-4) \times 3 = -24\).
• Addition and subtraction complete the simplification of the expression: \(144 - 24 + 4\).

Mastering these operations makes it easier to follow the evaluation process and ensure accurate results.