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Chapter 10: Problem 202

Multiply the following binomials. Use any method. $$(2 q-5)(7 q-11)$$

Short Answer

Expert verified
14q^2 - 57q + 55

Step by step solution

01

- Use the Distributive Property

Apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first binomial by each term in the second binomial. First: Multiply the first terms: (2q)(7q) = 14q^2 Outer: Multiply the outer terms: (2q)(-11) = -22q Inner: Multiply the inner terms: (-5)(7q) = -35q Last: Multiply the last terms: (-5)(-11) = 55
02

- Combine Like Terms

Combine the intermediate terms you obtained: 14q^2 - 22q - 35q + 55 Combine the terms with q: 14q^2 - 57q + 55

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in mathematics. It allows us to multiply a single term by two or more terms inside a set of parentheses. For binomials, this process is extended to each term within the first binomial to each term in the second binomial. In simpler terms, you distribute each part of the first binomial to every part of the second binomial. This can be broken down as follows for the example \( (2 q-5)(7 q-11) \):
  • \textbf{First:} Multiply the first terms: \( (2q)(7q) = 14q^2 \)
  • \textbf{Outer:} Multiply the outer terms: \( (2q)(-11) = -22q \)
  • \textbf{Inner:} Multiply the inner terms: \((-5)(7q) = -35q\)
  • \textbf{Last:} Multiply the last terms: \((-5)(-11) = 55 \)
This method ensures that every term is properly multiplied, setting the stage for the next step: combining like terms.
FOIL method
The FOIL method is a handy mnemonic for remembering the steps to multiply two binomials using the distributive property. FOIL stands for:
  • \textbf{F}irst
  • \textbf{O}uter
  • \textbf{I}nner
  • \textbf{L}ast

This method specifically applies to the multiplication of binomials. By following FOIL, you ensure that each term in the first binomial is multiplied by each term in the second binomial systematically. In our example \((2 q-5)(7 q-11)\), using the FOIL method gives us the same results:
  • First: \( (2q)(7q) = 14q^2\)
  • Outer: \( (2q)(-11) = -22q \)
  • Inner: \((-5)(7q) = -35q\)
  • Last: \((-5)(-11) = 55 \)

Thus, the FOIL method is essentially another way to apply the distributive property, but in an easy-to-remember sequence.
combining like terms
After using the distributive property (or the FOIL method), you'll be left with several terms that need to be simplified by combining like terms. Like terms are terms that have the same variables raised to the same power. In our example \((2 q-5)(7 q-11)\), after performing all multiplications, we get:
  • \(14q^2\)
  • \(-22q\)
  • -35q
  • 55

The next step is to combine the like terms for the variable \(q\). Here, both \(-22q\) and \(-35q\) are like terms:
  • 14q^2 - 22q - 35q + 55
Combining these gives:
  • 14q^2 - 57q + 55
Combining like terms helps simplify the expression, making it easier to understand and work with. Ensure you always look for and combine like terms to arrive at the simplest form of your polynomial.

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Most popular questions from this chapter

Multiply and write your answer in decimal form. $$\frac{6 \times 10^{9}}{2 \times 10^{-1}}$$

Factor the greatest common factor from each polynomial. $$5 y^{2}-55 y+45$$

Factor the greatest common factor from each polynomial. $$6 p^{2}+6 p$$

Simplify each expression. $$(n-8)\left(n^{2}-4 n+11\right)$$

Divide the monomials. $$72 p^{12} \div 8 p^{3}$$
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