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Chapter 5: Problem 29

Find each product. $$ (6 x+1)\left(2 x^{2}+4 x+1\right) $$

Short Answer

Expert verified
12x^3 + 26x^2 + 10x + 1

Step by step solution

01

- Apply the distributive property

Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. This means expanding \( (6x+1)(2x^2+4x+1) \) into individual terms: \[ 6x \times 2x^2 + 6x \times 4x + 6x \times 1 + 1 \times 2x^2 + 1 \times 4x + 1 \times 1 \]
02

- Perform the multiplications

Multiply the terms calculated in Step 1:\[ 6x \times 2x^2 = 12x^3 \]\[ 6x \times 4x = 24x^2 \]\[ 6x \times 1 = 6x \]\[ 1 \times 2x^2 = 2x^2 \]\[ 1 \times 4x = 4x \]\[ 1 \times 1 = 1 \]
03

- Combine like terms

Add together all the terms from Step 2, combining the like terms:\[ 12x^3 + 24x^2 + 2x^2 + 6x + 4x + 1 = 12x^3 + 26x^2 + 10x + 1 \]

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Key Concepts

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

distributive property
To start multiplying polynomials, we use the distributive property. This property helps us break down a complex multiplication problem into simpler parts.
Here's how we apply it to our exercise:

When we have two polynomials, like \( (6x + 1)(2x^2 + 4x + 1) \), we multiply each term in the first polynomial by every term in the second polynomial.
So, we expand \( (6x+1)(2x^2+4x+1) \) into individual terms:

  • 6x \times 2x^2
  • 6x \times 4x
  • 6x \times 1
  • 1 \times 2x^2
  • 1 \times 4x
  • 1 \times 1
This step helps us simplify the initial multiplication problem by breaking it down.
Next, we just multiply these individual pairs.
combining like terms
After using the distributive property to expand the polynomials, we perform the individual multiplications.
From our expanded terms, we get:

  • \(6x \times 2x^2 = 12x^3\)
  • \(6x \times 4x = 24x^2\)
  • \(6x \times 1 = 6x\)
  • \(1 \times 2x^2 = 2x^2\)
  • \(1 \times 4x = 4x\)
  • \(1 \times 1 = 1\)
Now, we have several terms. The next step is to combine the like terms.
Like terms are terms that include the same variable raised to the same power.
Here's our list of like terms:
  • 24x^2 and 2x^2
  • 6x and 4x
Combining these, our equation simplifies:

\( 12x^3 + 24x^2 + 2x^2 + 6x + 4x + 1 \)
Adding like terms results in:

\( 12x^3 + 26x^2 + 10x + 1 \)
This simplified form makes it easier to understand and work with the polynomial.
multiplying polynomials
Multiplying polynomials involves several key steps.
Let's revisit how we multiplied \( (6x + 1)(2x^2 + 4x + 1) \).

  • First, we distribute each term in the first polynomial across every term in the second polynomial.
  • Then, we perform each multiplication to get several new terms.
  • Finally, we combine like terms to simplify the expression.
Polynomials can have many terms, but by using these steps, we can systematically break down the multiplication process.

Let's use another example for clarity:
Imagine you need to multiply聽\( (x+2)(x-3) \).
Using the distributive property, distribute each term:
  • \(x \times x = x^2\)
  • \(x \times -3 = -3x\)
  • \(2 \times x = 2x\)
  • \(2 \times -3 = -6\)
Now, combining like terms:
\( x^2 - 3x + 2x - 6 \)
Which simplifies to:
\( x^2 - x - 6 \)
Practicing these steps will make polynomial multiplication much more manageable!

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