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Chapter 5: Problem 51

Find each product. Use the FOIL method. $$ (3 x-2)(3 x-2) $$

Short Answer

Expert verified
The product is: \[ 9x^2 - 12x + 4 \]

Step by step solution

01

- Identify the terms

Recognize that we have two binomials: \(3x - 2\) and \(3x - 2\).
02

- Apply the FOIL method (First)

Multiply the first terms in each binomial: \[ 3x \times 3x = 9x^2 \]
03

- Apply the FOIL method (Outer)

Multiply the outer terms in each binomial: \[ 3x \times -2 = -6x \]
04

- Apply the FOIL method (Inner)

Multiply the inner terms in each binomial: \[ -2 \times 3x = -6x \]
05

- Apply the FOIL method (Last)

Multiply the last terms in each binomial: \[ -2 \times -2 = 4 \]
06

- Combine like terms

Sum all the products obtained: \[ 9x^2 - 6x - 6x + 4 = 9x^2 - 12x + 4 \]

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Key Concepts

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

Understanding Binomials
A binomial is an algebraic expression containing exactly two terms.

These terms are usually separated by a plus (+) or minus (鈭�) sign.

For example, in the expression \(3x - 2\), there are two terms: \(3x\) and \(-2\).
This means it's a binomial.

Recognizing binomials is crucial when applying methods like FOIL because it tells you how many terms you'll work with.

Another example of a binomial is \(x + 5\).
Here, the two terms are \(x\) and \(5\).

When multiplying binomials, you always need to use each term from the first binomial with each term from the second binomial.
Multiplication of Polynomials
Multiplying polynomials involves distributing each term in the first polynomial to every term in the second polynomial.

In our example, we use the FOIL method, which stands for First, Outer, Inner, Last.

The FOIL method helps us remember the order in which to multiply terms:

  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in each binomial.
  • Inner: Multiply the inner terms in each binomial.
  • Last: Multiply the last terms in each binomial.

Let鈥檚 look at our example \( (3x - 2)(3x - 2) \).
When we apply the FOIL method:

  • First: \ (3x \times 3x = 9x^2) \
  • Outer: \ (3x \times -2 = -6x) \
  • Inner: \ (-2 \times 3x = -6x) \
  • Last: \ (-2 \times -2 = 4) \
Each product is calculated step-by-step to arrive at the final result.
Combining Like Terms
Combining like terms involves adding or subtracting terms that have the same variable raised to the same power.

This simplifies the expression.

After using the FOIL method, we obtain \ 9x^2 - 6x - 6x + 4 \.
Now we need to combine the like terms:

Here, \( -6x \) and \( -6x \) are like terms because they both have the variable \(x\):
\ -6x - 6x = -12x \.

This simplification gets the expression to its final form:
\ 9x^2 - 12x + 4 \.
Combining like terms helps in reducing the polynomial to its simplest form, making it easier to understand and work with.

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Most popular questions from this chapter

Evaluate. $$ 38 \div 10 $$

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