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Chapter 5: Problem 194

Multiply the binomials. Use any method. $$ (q-5)(q+8) $$

Short Answer

Expert verified
\text(q-5)(q+8) = q^2 + 3q - 40

Step by step solution

01

Write Down the Binomials

Identify the two binomials to be multiplied: \(q-5\) and \(q+8\).
02

Apply the Distributive Property (FOIL Method)

Use the FOIL method to expand the binomials. FOIL stands for First, Outer, Inner, Last: First: Multiply the first terms: \(q \cdot q = q^2\) Outer: Multiply the outer terms: \(q \cdot 8 = 8q\) Inner: Multiply the inner terms: \((-5) \cdot q = -5q\) Last: Multiply the last terms: \((-5) \cdot 8 = -40\)
03

Combine Like Terms

Combine the like terms from the expanded form: \(q^2 + 8q - 5q - 40\) Simplify this to get: \(q^2 + 3q - 40\)

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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 useful principle in algebra. It allows us to break down expressions to simplify and multiply them. The formula is:
  • For any numbers or variables: \( a ( b+c ) = ab + ac \)

For our problem, which is to multiply the binomials \(( q-5 )( q+8 ) \), we need to apply the distributive property.
To do this, we distribute each term in the first binomial to every term in the second binomial. This is where the FOIL method comes in handy.
FOIL method
The FOIL method is a helpful tool for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, and it reminds us of the order in which to multiply the terms. For our binomials \(( q-5 )( q+8 )\), we follow these steps:
  • First: Multiply the first terms: \( q \times q = q^2 \)
  • Outer: Multiply the outer terms: \( q \times 8 = 8q \)
  • Inner: Multiply the inner terms: \( -5 \times q = -5q \)
  • Last: Multiply the last terms: \( -5 \times 8 = -40 \)

Putting all of these together, we get: \( q^2 + 8q - 5q - 40 \). This gives us all the individual products we need.
combining like terms
After using the FOIL method, we have the expanded expression: \( q^2 + 8q - 5q - 40 \).
Our next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our example, \( 8q \) and \( -5q \) are like terms because they both have the variable \( q \):

By combining \( 8q - 5q \), we get:
\( 8q - 5q = 3q \).
Now, our simplified expression is: \( q^2 + 3q - 40 \).
This is our final answer after correctly distributing, applying the FOIL method, and combining like terms!

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Most popular questions from this chapter

Square each binomial using the Binomial Squares Pattern. $$ \left(y+\frac{1}{4}\right)^{2} $$

Square each binomial using the Binomial Squares Pattern. $$ \left(8 p^{3}-3\right)^{2} $$

Multiply the binomials using (a) the Distributive Property; (b) the FOIL method; (c) the Vertical Method. $$ (4 p+11)(5 p-4) $$

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Multiply using (a) the Distributive Property; (b) the Vertical Method. $$ (u+4)\left(u^{2}+3 u+2\right) $$
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