91Ó°ÊÓ

Combining like terms is a crucial part of polynomial operations that involves simplifying an expression by consolidating similar components. Like terms are terms that have the same variable raised to the same power. For example, in our given exercise, the terms \(3x^2\) and \(-2x^2\) are like terms because they both involve \(x^2\). Similarly, the terms \(x\) and \(3x\) both involve \(x\) raised to the first power, making them like terms as well.

By combining like terms, we add or subtract the coefficients while keeping the variable part unchanged. In the example \( (3x^2 + x + 1) - (2x^2 - 3x - 5) \), we can simplify it by combining \(3x^2\) and \(-2x^2\) to get \(x^2\). Similarly, combining \(x\) and \(3x\) results in \(4x\), and adding the constants \(1\) and \(5\) results in \(6\).

The simplified expression becomes \(x^2 + 4x + 6\). Combining like terms helps in arriving at a more concise expression, making it easier to interpret and solve further.

The distributive property is a fundamental concept in algebra and polynomial operations. It allows us to multiply a single term by each term inside a bracket. This is especially helpful when simplifying expressions that involve parentheses.

In our exercise, we initially have the expression \((3x^2 + x + 1) - (2x^2 - 3x - 5)\). The negative sign in front of the second set of parentheses is distributed to each term within the brackets. Consequently, \(-(2x^2 - 3x - 5)\) transforms to \(-2x^2 + 3x + 5\). This step is essential for correctly simplifying the expression because it ensures each term is accurately considered.

• The negative sign changes the sign of each term in the second bracket: \(2x^2\) turns into \(-2x^2\), \(-3x\) turns into \(+3x\), and \(-5\) turns into \(+5\).
• Distributing ensures that addition or subtraction can be performed correctly with like terms like we saw in the previous section.

Polynomial simplification is a process of making an expression easier to work with by reducing it to its simplest form. It involves several steps, including applying the distributive property and combining like terms.

Starting with the expression \((3x^2 + x + 1) - (2x^2 - 3x - 5)\), we used the distributive property to remove the parentheses, resulting in \(3x^2 + x + 1 - 2x^2 + 3x + 5\). From there, combining like terms allows us to streamline the equation to \(x^2 + 4x + 6\).

• We simplified \(3x^2 - 2x^2\) to get \(x^2\).
• Adding the \(x\) terms, \(x + 3x\), gave us \(4x\).
• Finally, the constant terms \(1 + 5\) simplify to \(6\).

This whole process sharpens the polynomial expression, making it simpler and ready for further analyses or solutions. Simplification reduces potential confusion and helps in better understanding the structure and value of the polynomial.