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Chapter 5: Problem 22

Multiplying Any Two Polynomials Multiply. $$(3 r-4)(2 r+1)$$

Short Answer

Expert verified
6r^2 - 5r - 4

Step by step solution

01

Distribute Each Term in the First Polynomial

Distribute each term in the polynomial (3r - 4)multiply each term by each term in the polynomial (2r + 1). This gives us four smaller products: (3r)(2r), (3r)(1), (-4)(2r), and (-4)(1).
02

Multiply Each Pair of Terms

Now, multiply each pair of terms obtained in Step 1: (3r)(2r) = 6r^2, (3r)(1) = 3r, (-4)(2r) = -8r, and (-4)(1) = -4.
03

Combine Like Terms

Combine the linear terms from the products: 3r and -8r. This gives a single linear term: 3r - 8r = -5r.
04

Write Final Expression

Finally, combine all terms to write the expression representing the product of the polynomials. The polynomial is: 6r^2 - 5r - 4.

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Key Concepts

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

distributive property
In this section, let's explore the distributive property. This property states that for any three numbers, a, b, and c: a(b + c) = ab + acThis means you can distribute a multiplication over an addition. In the polynomial multiplication (3r - 4)(2r + 1), we use the distributive property to break down the multiplication process.
First, we distribute 3r to both terms inside the parentheses (2r + 1), giving us:
  • (3r)(2r) = 6r^2
  • (3r)(1) = 3r
Next, we distribute -4 to both terms inside the parentheses (2r + 1), giving us:
  • (-4)(2r) = -8r
  • (-4)(1) = -4
This step-by-step application of the distributive property helps break the multiplication of two polynomials into simpler, manageable parts.
combining like terms
Combining like terms is essential after applying the distributive property. Like terms are terms that have the same variable raised to the same power. For example, in our case, 3r and -8r are like terms because they both have the variable r raised to the first power.
After applying the distributive property, we have these terms:
  • 6r^2
  • 3r
  • -8r
  • -4
The next step is to combine 3r and -8r. We do this by adding or subtracting their coefficients (the numbers in front of the variables). So, 3r - 8r = -5r.
Now, we can write the expression with our combined like terms: 6r^2 - 5r - 4.
polynomial multiplication
Polynomial multiplication simply means multiplying two expressions that contain variables raised to a power. In our example, we multiply (3r - 4) by (2r + 1). This involves both the distributive property and combining like terms.

Let's summarize the steps:
1. Distribute each term in the first polynomial to each term in the second polynomial.
2. Multiply the corresponding pairs of terms.
3. Combine like terms to simplify the expression.
So, multiplying the terms in our expression gives:
  • (3r)(2r) = 6r^2
  • (3r)(1) = 3r
  • (-4)(2r) = -8r
  • (-4)(1) = -4
After multiplying, we combine the like terms 3r and -8r to get -5r. Thus, the product of the polynomials is:
6r^2 - 5r - 4.
This method ensures organized and accurate polynomial multiplication.

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