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Expanding binomials often begins with using the distributive property. This property lets you distribute or spread out multiplication over addition or subtraction within parentheses. Consider the expression

• \((x+5)(x+2)\). Here, the distributive property allows us to separate the expressions into smaller, more manageable pieces.

The operation can be broken down as:

• Distribute the first term in the first binomial to both terms in the second binomial
• Do the same with the second term in the first binomial
• Combine the results together

This gives us:\[(x+5)(x+2) = x(x+2) + 5(x+2)\]Observe how each term from the first binomial is distributed to every term in the second one. This step is crucial before moving on to a more specific method like FOIL.

The FOIL method is a specific application of the distributive property for binomials, standing for First, Outer, Inner, Last. It helps you remember the order in which to multiply the terms in two binomials. Here's what each part means:

• First: Multiply the first terms in each binomial: \(x \times x = x^2\)
• Outer: Multiply the outer terms: \(x \times 2 = 2x\)
• Inner: Multiply the inner terms: \(5 \times x = 5x\)
• Last: Multiply the last terms in each binomial: \(5 \times 2 = 10\)

By applying the FOIL method to our expression, we get:\[x^2 + 2x + 5x + 10\]This structured approach simplifies the task of expanding binomials by handling it in a step-by-step manner. Remember, though FOIL is very handy, it's specifically tailored for binomials and doesn't apply to trinomials or other types of polynomials.

Once you have expanded the binomials using either the distributive property or the FOIL method, the next step is to combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. This is essential, as it simplifies the expression, making it more convenient to use. Let's take a look at our example:

• The expanded expression is\(x^2 + 2x + 5x + 10\)
• Here, \(2x\) and \(5x\) are like terms because they both contain the variable \(x\) raised to the first power

To combine them, simply add their coefficients:\(2x + 5x = 7x\)So, the simplified expression becomes:\[x^2 + 7x + 10\]Remember, combining like terms is a crucial step in polynomial operations, as it ensures that the expression is in its simplest and most efficient form. Always look for and combine like terms to achieve a clean and concise result.