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Chapter 4: Problem 36

Use the FOIL method to find each product. $$ (p+4)(p-6) $$

Short Answer

Expert verified
\( p^2 - 2p - 24 \)

Step by step solution

01

Identify the terms

Identify the four terms in the binomials. The terms are: First binomial: \(p + 4\) Second binomial: \(p - 6\)
02

Apply the FOIL method

FOIL stands for First, Outer, Inner, Last. First: Multiply the first terms from each binomial: \( p \cdot p = p^2 \) Outer: Multiply the outer terms from each binomial: \( p \cdot (-6) = -6p \) Inner: Multiply the inner terms from each binomial: \( 4 \cdot p = 4p \) Last: Multiply the last terms from each binomial: \( 4 \cdot (-6) = -24 \)
03

Combine like terms

Combine the products from the FOIL method: \( p^2 - 6p + 4p - 24 \). Simplify by combining like terms: \( p^2 - 2p - 24 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomials
A binomial is a polynomial with exactly two terms. These terms can be variables, constants, or a combination of both. For example, in the expression \(p + 4\), \(p\) is a variable term and \(4\) is a constant term.

Binomials are typically represented in the form \(a + b\) where \(a\) and \(b\) are any terms. Understanding the structure of binomials is crucial for operations like addition, subtraction, and multiplication.
In our exercise, we are working with two binomials: \(p + 4\) and \(p - 6\). Each binomial comprises two parts, making it straightforward for us to apply methods such as the FOIL method to multiply them.
Polynomial Multiplication
Polynomial multiplication involves multiplying each term in the first polynomial by each term in the second polynomial. This can often be simplified using methods like the FOIL method, especially when dealing with binomials.

In our exercise, we use the FOIL method, which stands for:
  • First: Multiply the first terms from each binomial.
  • Outer: Multiply the outer terms in the binomials.
  • Inner: Multiply the inner terms in the binomials.
  • Last: Multiply the last terms in the binomials.
This method helps break down the multiplication process into manageable steps.
Let's look at our problem step by step:
First: \(p \times p = p^2 \)
Outer: \(p \times (-6) = -6p\)
Inner: \(4 \times p = 4p\)
Last: \(4 \times (-6) = -24\)
These steps collectively simplify polynomial multiplication by systematically addressing each term.
Combining Like Terms
Combining like terms is the process of simplifying a polynomial by adding or subtracting terms that have the same variable raised to the same power. This step is critical to getting the final simplified form of your polynomial.

In our exercise, after applying the FOIL method, we get the expression \(p^2 - 6p + 4p - 24\). To simplify this, we need to combine the like terms:

  • The terms \(-6p\) and \(4p\) both have the variable \(p\). We combine them to get \(-2p\).
So the simplified expression is: \(p^2 - 2p - 24\)
This step ensures that our polynomial is in its simplest form and is easier to work with for further mathematical operations or evaluations.

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Most popular questions from this chapter

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ (0.2 x-1.4 y)^{2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(-5 y^{3} z^{4}\right)^{2}\left(2 y z^{5}\right)^{-2}}{10\left(y^{4} z\right)^{3}\left(3 y^{3} z^{2}\right)^{-1}} $$

Perform the indicated operations. $$ 5 k-(5 k-[2 k-(4 k-8 k)])+11 k-(9 k-12 k) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ k^{-5} k^{-3} k^{4} $$

Find each product. $$ (q-2)^{4} $$
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