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Multiplying binomials involves applying the distributive property to expand the expression. When you see an expression like \((a+b)(c+d)\), think of it as multiplying each part of the first binomial with every part of the second. This is sometimes called the "FOIL" method, which stands for First, Outer, Inner, Last. However, this method only applies directly to two binomials.To illustrate, if we look at the provided expression \( (x^2-2x+1)(x-5) \), each term in the first binomial \((x^2, -2x, 1)\) is multiplied by each term in the second \((x, -5)\):

• First multiply \(x^2\) by \(x\), giving \(x^3\).
• Next, \(x^2\) times \(-5\) results in \(-5x^2\).
• Then, \(-2x\) times \(x\) which is \(-2x^2\).
• Next, \(-2x\) times \(-5\), which gives \(10x\).
• Then, \(1\) times \(x\), giving \(x\).
• Finally, \(1\) times \(-5\), resulting in \(-5\).

Once expanded, you have the individual terms\(x^3 - 5x^2 - 2x^2 + 10x + x - 5\). The next step is to simplify by combining like terms.

After expanding the binomials, the resultant expression often includes like terms. Like terms are terms that have the same variable raised to the same power. When you combine them, you simply add or subtract their coefficients.In the expression \(x^3 - 5x^2 - 2x^2 + 10x + x - 5\), let's break down how you find and combine the like terms:

• Look for terms with the same variable and exponent. Here, \(-5x^2\) and \(-2x^2\) are like terms because they both have \(x^2\).
• Combine \(-5x^2\) and \(-2x^2\) to get \(-7x^2\).
• The terms \(10x\) and \(x\) are also like terms, both with \(x\) as the variable.
• Combine \(10x\) and \(x\) to get \(11x\).

Now, you have a new, simplified polynomial: \(x^3 - 7x^2 + 11x - 5\). This makes it easier to analyze or graph if needed.

Simplifying expressions is the process of making a polynomial as uncomplicated as possible by combining like terms and performing arithmetic operations.Let's consider our result \(x^3 - 7x^2 + 11x - 5\). Each term here is simplified because it has been combined with other like terms, and all arithmetic has been executed.

• Ensure each variable term has its coefficient clearly shown. In this case, terms \(x^3\), \(-7x^2\), and \(11x\) are neatly aligned with their coefficients.
• Double-check that there are no further operations you can perform. This means your polynomial is as easy to work with as possible.
• Recognize that sometimes simplifying helps to solve further mathematical questions, such as finding roots or intercepts.

Simplification is essential for efficient communication of mathematical ideas. A tidy polynomial like \(x^3 - 7x^2 + 11x - 5\) allows for greater ease in further calculations and problem-solving.