91Ó°ÊÓ

Polynomial expansion involves applying the Binomial Theorem to express a binomial raised to a power as a sum of terms. Each term in this expansion is made up of coefficients, variables, and exponents. The Binomial Theorem provides a systematic method for expanding polynomials and says that any binomial expression \[ (a+b)^n \] can be expanded as: \[ \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \] where \( \binom{n}{k} \) is a binomial coefficient that can be calculated as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]In our exercise, - the expression \((x-3)^4\) requires finding each term using binomial coefficients and powers of both \(x\) and \(-3\): - First term: \(x^4\) - Second term: \(-12x^3\) - Third term: \(54x^2\) - Fourth term: \(-108x\) - Fifth term: \(81\). - Similarly, for \((x-3)^2\): - First term: \(x^2\) - Second term: \(-6x\) - Third term: \(9\).
Each term results from applying the coefficients and combining powers of \(x\) and \(-3\).
Expanding polynomials helps to simplify expressions and solve problems.

Once the binomial expression is expanded, simplification is crucial to make it understandable and workable. Simplifying involves combining like terms and carrying out basic arithmetic such as addition and subtraction. In the example problem, the expansion of \((x-3)^4\) gives: - \(x^4\) - \(-12x^3\) - \(54x^2\) - \(-108x\) - \(81\)and the expansion of \((x-3)^2\) results in: - \(x^2\) - \(-6x\) - \(9\).
After expanding, you should focus on each part:

• Identify like terms.
• Combine the like terms by adding or subtracting them.

Usually, terms are organized in descending order of their exponents for neatness and simplicity.
For example, all terms with the same power of \(x\), such as \(x^2\) terms, are added together, making the expression simpler. Simplification also often prepares expressions for further steps in solving equations or substituting them back into other expressions.

The distributive property is a foundational algebraic principle that states: \[ a(b+c) = ab + ac \] It allows you to distribute a factor across terms inside parentheses, effectively breaking down complex expressions into simpler parts. This property is essential in both expansion and simplification processes. In our exercise:

• The expanded form of \( 2(x-3)^4 \) requires distributing the 2 across all terms of the \((x-3)^4\) expansion.
• Similarly, the 5 must be distributed across the \((x-3)^2\) terms.

This means you multiply 2 by each term in \((x^4 - 12x^3 + 54x^2 - 108x + 81)\) resulting in:

• \(2x^4\)
• - \(24x^3\)
• + \(108x^2\)
• - \(216x\)
• + \(162\)

Similarly, multiply 5 by each term in \((x^2 - 6x + 9)\) to get:

• \(5x^2\)
• - \(30x\)
• + \(45\)

This distributes and expands all parts of the expression. Afterwards, you combine these into a single expression, maintaining clarity and simplicity in the final result:
\[ 2x^4 - 24x^3 + 113x^2 - 246x + 207 \]
Applying the distributive property efficiently enables problem-solving and simplification in tasks involving algebraic expressions.