91Ó°ÊÓ

The Distributive Property is a key tool in algebra that allows us to multiply a single term across terms inside parentheses. This principle states that multiplying a sum (or difference) by a number gives the same result as multiplying each summand separately and then adding (or subtracting) the products. In essence, it breaks down more complicated expressions into simpler parts.

For example, let's simplify the expression \(4(3x^2-5x+2)\). Here, the number 4 outside the parentheses will be multiplied by each term inside the parentheses. We do this for every term:

• \(4\times(3x^2) = 12x^2\)
• \(4\times(-5x) = -20x\)
• \(4\times(2) = 8\)

Repeating this process with \(3(2x^2+6x-3)\) results in:

• \(3\times(2x^2) = 6x^2\)
• \(3\times(6x) = 18x\)
• \(3\times(-3)= -9\)

Breaking down expressions into these simpler components helps when moving on to combining like terms.

After using the distributive property, the next step in algebraic simplification is combining like terms. Like terms are terms that have the same variable raised to the same power. Only like terms can be combined to simplify an expression further.

In our example, after distributing the terms, we have the polynomial:

• \(12x^2 - 20x + 8 + 6x^2 + 18x - 9\)

We can see several like terms here:

• Terms with \(x^2\) are \(12x^2\) and \(6x^2\).
• Terms with \(x\) are \(-20x\) and \(18x\).
• Constants are \(8\) and \(-9\).

When combining, simply add or subtract the coefficients of these like terms:

• \(12x^2 + 6x^2 = 18x^2\)
• \(-20x + 18x = -2x\)
• \(8 - 9 = -1\)

This step leaves us with a simplified polynomial expression: \(18x^2 - 2x - 1\). Combining like terms is crucial for reducing expressions to their simplest form.

Polynomial expressions are algebraic expressions that consist of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental in algebra as they provide a standardized way to represent mathematical relationships.

In its simplest form, a polynomial might be a single-term polynomial, or monomial. When composed of two terms, it becomes a binomial, and with three, a trinomial. Polynomials can extend to any number of terms.

In our given problem, the expression \(4(3x^2-5x+2)+3(2x^2+6x-3)\) is a polynomial expression. Each part of this expression has been expanded and simplified, showing that it was originally composed of two trinomials. After applying the distributive property and combining like terms, this expression simplifies to \(18x^2 - 2x - 1\), which itself is a trinomial.

Recognizing and working with polynomial expressions is crucial for anyone studying algebra, as they provide the basis for more complex mathematical concepts and operations.