91Ó°ÊÓ

Polynomials are algebraic expressions that consist of variables and coefficients, connected by addition, subtraction, and multiplication. A single term like \(3x\) or \(-2x^2\) is called a monomial. When you combine two or more monomials, you form a polynomial. Polynomials are an essential part of algebra because they allow for a wide range of mathematical operations and manipulations.
In the given exercise, we are working with the polynomial \(3x(-2x+5)\). Understanding polynomials helps us distribute, simplify, and solve complex algebraic expressions.

Multiplying variables is a crucial skill in algebra. When you multiply variables, you combine their exponents. For example, \(x \times x = x^2\). In the exercise, we first multiplied \(3x\) by \(-2x\). Here, the constants \(3\) and \(-2\) are multiplied to get \(-6\), and the variable \(x \times x\) yields \(x^2\), resulting in \(-6x^2\).
Next, we multiplied \(3x\) by 5. multiplying constants gives \(3 \times 5 = 15\), and we carry along the variable \(x\), resulting in \(15x\). Multiplying variables this way allows us to simplify expressions into a more manageable form.

Algebraic expressions are mathematical phrases that include numbers, variables, and arithmetic operations. They can sometimes include parentheses to indicate which operations should be performed first, just like in our given expression \(3x(-2x+5)\).
To solve such expressions, you often use the distributive property, which lets you multiply a single term across terms inside parentheses. This is crucial for expanding and simplifying expressions. In our exercise, applying the distributive property gave us the terms \(-6x^2\) and \(15x\), which we then combined to form the simplified algebraic expression \(-6x^2 + 15x\).
This process is fundamental in algebra because it helps in breaking down complex problems into simpler, understandable steps.