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Chapter 5: Problem 193

Multiply the binomials. Use any method. $$ (6 p+5)(p+1) $$

Short Answer

Expert verified
The product is \(6p^2 + 11p + 5 \).

Step by step solution

01

Use the Distributive Property

To multiply the binomials \( (6p+5)(p+1) \), use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first binomial by each term in the second binomial.
02

Multiply the First Terms

Multiply the first terms in each binomial: \(6p \cdot p = 6p^2 \).
03

Multiply the Outer Terms

Multiply the outer terms in the binomials: \(6p \cdot 1 = 6p \).
04

Multiply the Inner Terms

Multiply the inner terms in the binomials: \(5 \cdot p = 5p \).
05

Multiply the Last Terms

Multiply the last terms in the binomials: \(5 \cdot 1 = 5 \).
06

Combine Like Terms

Add all the products together to get: \(6p^2 + 6p + 5p + 5 \). Combine the like terms \(6p \) and \(5p \) to get the final simplified expression: \(6p^2 + 11p + 5 \).

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Key Concepts

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

distributive property
Multiplying binomials involves using a concept called the distributive property. This property helps in breaking down more complex problems by spreading or distributing one term over the others. Imagine you have two sets of objects and you want to pair each object of the first set with every object of the second set. That's what we do when multiplying binomials. In mathematical terms, if you have two binomials \((a+b)(c+d)\), you distribute the terms like this: \(a(c+d) + b(c+d)\), resulting in \(ac + ad + bc + bd\). This method makes it easier to see and handle each part separately before combining them again.
FOIL method
An application of the distributive property when multiplying binomials is known as the FOIL method. FOIL stands for:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms.
Using our example \((6p+5)(p+1)\), we apply the FOIL method as follows:
1. First: \(6p \cdot p = 6p^2\).
2. Outer: \(6p \cdot 1 = 6p\).
3. Inner: \(5 \cdot p = 5p\).
4. Last: \(5 \cdot 1 = 5\).
After multiplying each pair of terms, you add the results together: \(6p^2 + 6p + 5p + 5\).
combining like terms
The last step involves combining like terms to simplify the expression. 'Like terms' are terms that have the same variable raised to the same power. In our example, the terms \(6p\) and \(5p\) are like terms.
Add them together: \(6p + 5p = 11p\). The final simplified expression is: \(6p^2 + 11p + 5\). Combining like terms helps to consolidate the expression, making it easier to understand and work with further.

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