91Ó°ÊÓ

Polynomial distribution, also known as the distributive property, is a fundamental concept in algebra. It involves multiplying a term outside a set of parentheses with each term inside the parentheses. In our exercise, we started with the expression \(9t^4 + 7t - 5(9t^3 - 2t)\).
The distribution step required us to multiply \(-5\) by both \(9t^3\) and \(-2t\). This means:
- Multiply \(-5\) by \(9t^3\): \(-5 \times 9t^3 = -45t^3\)
- Multiply \(-5\) by \(-2t\): \(-5 \times -2t = 10t\)
This results in the new terms: \(-45t^3 + 10t\). Combining these new terms back into the original expression results in: \(9t^4 + 7t - 45t^3 + 10t\). The purpose of this step is to eliminate the parentheses, making it easier to manage and simplify the expression in subsequent steps.

Combining like terms helps to simplify expressions by merging terms with identical variables and exponents. In our example, after distributing, we have the expression \(9t^4 + 7t - 45t^3 + 10t\).
Here, 'like terms' are terms with the same variables raised to the same power.
- Identify like terms: \(7t\) and \(10t\) are like terms.
- Combine them: Add their coefficients (the numbers in front of the variables): \(7t + 10t = 17t\).
This modifies our expression to: \(9t^4 - 45t^3 + 17t\). Combining like terms reduces the expression to fewer, simpler terms, making calculations easier.

Polynomial simplification involves combining the distribution and like terms to achieve a simpler form of the polynomial. This makes it easier to handle and solve.
Let's review the overall process used in our example:
- Original expression: \(9t^4 + 7t - 5(9t^3 - 2t)\).
- First, we distributed \(-5\) across the terms inside the parentheses, transforming \(-5(9t^3 - 2t)\) into: \(-45t^3 + 10t\).
- After distribution, the expression was: \(9t^4 + 7t - 45t^3 + 10t\).
- Then, we combined like terms: \(7t + 10t = 17t\).
Final simplified expression: \(9t^4 - 45t^3 + 17t\). Simplification often requires multiple steps to systematically reduce the polynomial to its simplest form. This helps in further algebraic manipulations like solving equations, factoring, and more.