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Chapter 10: Problem 553

Multiply the binomials using various methods. $$(k+6)(k-9)$$

Short Answer

Expert verified
The product is \(k^2 - 3k - 54\).

Step by step solution

01

Apply the FOIL Method

FOIL stands for First, Outer, Inner, Last. Multiply the first terms, the outer terms, the inner terms, and the last terms together.First terms: \(k \times k = k^2\)Outer terms: \(k \times -9 = -9k\)Inner terms: \(6 \times k = 6k\)Last terms: \(6 \times -9 = -54\)
02

Combine Like Terms

Combine the terms from the FOIL method:\(k^2 - 9k + 6k - 54\)Combine the like terms (the coefficients of \(k\)):\(-9k + 6k = -3k\)So, the expression becomes: \(k^2 - 3k - 54\)
03

Write the Final Expression

The final expression after multiplying the binomials is:\(k^2 - 3k - 54\)

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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 special technique used for multiplying binomials. The acronym 'FOIL' stands for First, Outer, Inner, Last.
Here's how it works:

  • The 'First' terms are the first terms in each binomial. For \( (k+6)(k-9) \), these are \( k \ and \ k \), so \( k \times k = k^2 \).
  • The 'Outer' terms are the outermost terms in the multiplication. For \ (k+6)(k-9)\, these are \( k \ and \-9 \), giving \ k \times -9 = -9k \.
  • The 'Inner' terms are the innermost terms of the binomials. These are \ 6 \ and \ k \, resulting in \ 6 \times k = 6k \.
  • The 'Last' terms are the last terms of each binomial. So for \ (k+6)(k-9) \, it鈥檚 \ 6 \ and \ -9 \, leading to \ 6 \times -9 = -54 \.

Using the FOIL method ensures no part of the binomials is overlooked in the multiplication process.
Like Terms
Like terms are terms that have identical variable parts raised to the same power. When simplifying or combining terms, only like terms can be combined. This is crucial in binomial multiplication.
For the multiplication of \( (k+6)(k-9) \):

  • From FOIL, we have \ k^2, -9k, 6k, \ and \ -54 \.
  • Remember: \ k^2 \ is a quadratic term and cannot be combined with linear terms like \ -9k \ or \ 6k \.
  • The linear terms are \ -9k \ and \ 6k \. Since both terms have the variable \ k \, they are like terms and can be combined.

When we combine \ -9k \ and \ 6k \, we get \ -3k \ because \ -9+6 = -3 \. This simplifies our expression to \ k^2 - 3k - 54 \.
Binomial Multiplication
Binomial multiplication involves multiplying two binomials. A binomial is a polynomial with exactly two terms, like \( k+6 \) or \( k-9 \). To multiply them, the FOIL method is commonly used.
Here鈥檚 how you multiply \( (k+6)(k-9) \):

  • First, use the FOIL method to expand the expression: \ k^2, -9k, 6k, \ and \ -54 \.
  • Combine the like terms \ -9k \ and \ 6k \ to get \ -3k \.
  • Write everything together for the final expression: \ k^2 - 3k - 54 \.

The result of multiplying two binomials is a trinomial (a polynomial with three terms) in many cases.

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Most popular questions from this chapter

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