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Chapter 10: Problem 48

Explain why FOIL will not work when you multiply \((y+7)\left(y^{2}-3 y+1\right)\).

Short Answer

Expert verified
FOIL only works for two binomials, not a binomial and a trinomial. Use the distributive property instead.

Step by step solution

01

- Understand FOIL Method

The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, representing the terms you multiply together. However, this method only works for binomials (expressions with exactly two terms).
02

- Identify the Types of Expressions

Identify the expressions in the given problem. Here, \( y + 7 \) is a binomial because it has two terms. However, \( y^2 - 3y + 1 \) is a trinomial as it has three terms.
03

- Recognize the Incompatibility

Since FOIL is specifically designed for two binomials, it will not work for a binomial multiplied by a trinomial. The method is limited to expressions with exactly two terms each.
04

- Consider Alternative Methods

To multiply a binomial by a trinomial, use the distributive property instead. Distribute each term in the binomial across each term in the trinomial and then combine like terms.

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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 strategy specifically designed for multiplying two binomials. The abbreviation stands for First, Outer, Inner, Last. These represent the four pairs of terms to be multiplied together.
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the binomials.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms.

Remember, the FOIL method only works for binomials, not for expressions with more than two terms.
Binomial
A binomial is an algebraic expression that consists of exactly two terms. For example, \( a + b \) and \( y - 3 \) are binomials. When multiplying binomials, each term of the first binomial gets multiplied with each term of the second binomial. This results in four products, which can then be added together to form the final expression.
This is where the FOIL method is extremely useful, as it helps in organizing the multiplication of the two-term expressions.
Trinomial
A trinomial is an algebraic expression that consists of exactly three terms, such as \( y^2 - 3y + 1 \). Due to the extra term, trinomial multiplication is more complex than binomial multiplication. Here, the FOIL method is insufficient because FOIL applies only to two-term binomials.
When a binomial is multiplied by a trinomial, the distributive property is employed to handle the multiplication correctly.
Distributive Property
The distributive property is a useful algebraic property that helps in multiplying more complex expressions, like a binomial and a trinomial. It states that \( a(b + c) = ab + ac \). For a binomial and a trinomial, each term in the binomial is distributed across every term in the trinomial.
In our example, to multiply \( (y + 7)(y^2 - 3y + 1) \),
  • Distribute \( y \) across all terms in \((y^2 - 3y + 1)\)
  • Distribute \( 7 \) across all terms in \((y^2 - 3y + 1)\)
  • Combine all like terms.
This approach ensures all the terms are accounted for, making the multiplication accurate.

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