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Chapter 10: Problem 1

How do the letters in "FOIL" help you remember how to multiply two binomials?

Short Answer

Expert verified
The letters in 'FOIL' stand for First, Outer, Inner, and Last. These phrases correspond to the order in which terms of two binomials should be multiplied: First terms of both, the Outer terms in the product, the Inner terms, and finally the Last terms of both binomials. The FOIL method helps in quickly expanding the product of two binomials.

Step by step solution

01

Explanation of FOIL

FOIL is a mnemonic device used to remember the appropriate process for multiplying two binomials. Here, 'F' stands for First, 'O' for Outer, 'I' for Inner and 'L' for Last. This letter acronyms correspond to the terms in the binomial expressions that are being multiplied together.
02

Apply 'F' in FOIL to binomials

Take any two binomials such as \( (a+b) \) and \( (c+d) \). Applying 'F' (first) means multiplying the first terms of both binomials. In our specific case, the multiplication of the first terms is \( a \cdot c \).
03

Apply 'O' in FOIL to binomials

Next, applying 'O' (outer) means multiplying the outermost terms of the binomial. In this case, the multiplication of the outer terms is \( a \cdot d \).
04

Apply 'I' in FOIL to binomials

The next step is to apply 'I' (inner) that refers to the multiplication of the innermost terms of the binomial. Here, that multiplication is \( b \cdot c \).
05

Apply 'L' in FOIL to binomials

Lastly, applying 'L' (last) means multiplying the last terms of both binomials. Now, that multiplication is \( b \cdot d \). Now, add up all these four results to get the answer.
06

Final Result

\( (a+b) \cdot (c+d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d \). This is the expansion of the product of two binomials using FOIL.

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Key Concepts

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

FOIL method
The FOIL method is a simple way to multiply two binomials by following a specific sequence: First, Outer, Inner, Last. This mnemonic helps you identify which parts of the binomials you should multiply together. For example, consider two binomials, like \( (x+2) \) and \( (y+3) \):
  • First: Multiply the first terms, \( x \) and \( y \), resulting in \( xy \).
  • Outer: Multiply the outer terms, \( x \) and \( 3 \), yielding \( 3x \).
  • Inner: Multiply the inner terms, \( 2 \) and \( y \), giving you \( 2y \).
  • Last: Multiply the last terms, \( 2 \) and \( 3 \), resulting in \( 6 \).
After multiplying these pairs, you add them together: \( xy + 3x + 2y + 6 \). This approach ensures that nothing is left out and you get the correct expanded form of the binomial product. Using the FOIL method simplifies the process of multiplying binomials, making it more approachable.
Algebra
Algebra is an area of mathematics that uses symbols, letters, and numbers to represent and solve problems. It's like a puzzle where you work with unknowns and constants to find solutions. In the context of multiplying binomials, algebra helps us to simplify and manage the product of two binomials into a single expression.
When dealing with binomials, each term and operation plays a critical role. Algebra allows us to:
  • Identify and organize terms systematically, using structures like the FOIL method.
  • Combine like terms to simplify expressions further after using FOIL.
  • Use equations to solve for unknowns once binomials are expanded and simplified.
By learning algebra, students gain powerful tools to tackle a wide range of mathematical problems. It builds a foundational skill set that applies to topics such as solving equations, understanding functions, and analyzing more complex mathematical structures.
Binomials
A binomial is a specific type of polynomial composed of two terms. These terms are usually connected by addition or subtraction. For example, \( (a + b) \) and \( (x - y) \) are both binomials. The concept is crucial because binomials are the building blocks of more complex polynomial expressions.
Working with binomials involves:
  • Recognizing and identifying them within larger expressions.
  • Understanding how to perform operations, like the FOIL method, to expand or simplify them.
  • Utilizing binomials in equations to form logical arguments and solve problems.
Binomials appear frequently in algebra, making them a vital concept to understand. Their properties and operations, such as ones involved in the FOIL method, demonstrate the elegance and systematic organization of algebraic thought. This understanding also paves the way for more advanced topics in mathematics.

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Most popular questions from this chapter

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