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Chapter 5: Problem 15

Find each product. $$ (2 t+3)\left(3 t^{2}-4 t-1\right) $$

Short Answer

Expert verified
The product is \6t^{3} + t^{2} - 14t - 3\.

Step by step solution

01

- Distribute Each Term

Distribute each term in the first polynomial \(2t + 3\) to each term in the second polynomial \left( 3t^{2} - 4t - 1 \right)\. This means we'll need to multiply \(2t\) by every term in the second polynomial and then multiply \(3\) by every term in the second polynomial.
02

- Multiply the First Term

First, multiply \(2t\) by each term in the second polynomial: \[ 2t \times 3t^{2} = 6t^{3} \] \[ 2t \times -4t = -8t^{2} \] \[ 2t \times -1 = -2t \]
03

- Multiply the Second Term

Next, multiply \(3\) by each term in the second polynomial: \[ 3 \times 3t^{2} = 9t^{2} \] \[ 3 \times -4t = -12t \] \[ 3 \times -1 = -3 \]
04

- Combine Like Terms

Combine all like terms from the two sets of products: \[ 6t^{3} + (-8t^{2} + 9t^{2}) + (-2t - 12t) - 3 \] Simplifying this, we get: \[ 6t^{3} + t^{2} - 14t - 3 \]

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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 rule in algebra that allows us to multiply a single term by two or more terms inside a parenthesis. In essence, it states that \[a(b + c) = ab + ac\] In the context of our example, we need to apply this property to the polynomial \((2t + 3)(3t^2 - 4t - 1)\). Each term in \(2t + 3\) should be multiplied by each term in \(3t^2 - 4t - 1\). This means:
  • First, multiply \(2t\) by every term in the second polynomial:
    • \(2t \times 3t^2 = 6t^3\)
    • \(2t \times (-4t) = -8t^2\)
    • \(2t \times (-1) = -2t\)
  • Next, multiply \(3\) by every term in the second polynomial:
    • \(3 \times 3t^2 = 9t^2\)
    • \(3 \times (-4t) = -12t\)
    • \(3 \times (-1) = -3\)
After applying the distributive property, you'll have a series of terms that can be combined in the next step.
combining like terms
Once you have distributed each term, you will end up with several terms that need to be simplified. Combining like terms is a process where you add or subtract terms that have the same variable raised to the same power. In our example: \[6t^3 - 8t^2 - 2t + 9t^2 - 12t - 3\]We notice that we have multiple terms with the same variable:
  • Combining the \(t^2\) terms: \(-8t^2 + 9t^2 = t^2\)
  • Combining the \(t\) terms: \(-2t - 12t = -14t\)
So, our expression simplifies to: \[6t^3 + t^2 - 14t - 3\]This ensures that all our like terms are combined and we have a simplified polynomial.
degree of polynomial
The degree of a polynomial is determined by the highest power of the variable in the polynomial. It indicates the complexity and the number of solutions a polynomial equation could potentially have. For our resulting polynomial: \[6t^3 + t^2 - 14t - 3\]The term with the highest exponent is \(6t^3\), which means the degree of the polynomial is 3. To summarize, when determining the degree:
  • Identify the term with the highest power of the variable.
  • The exponent of this term is the degree of the polynomial.
In practical applications, knowing the degree of a polynomial helps in understanding its behavior, such as the number of possible roots and the general shape of its graph.

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Most popular questions from this chapter

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