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Chapter 6: Problem 2

Explain how to multiply a monomial by a trinomial.

Short Answer

Expert verified
To multiply a monomial by a trinomial, distribute the monomial term across each term in the trinomial and simplify the result by combining like terms. For example, if we have a monomial \(2x^2\) and a trinomial \(x^2 - 3x + 1\), we perform the following steps: \(2x^2 * x^2 = 2x^4\); \(2x^2 * (-3x) = -6x^3\); \(2x^2 * 1 = 2x^2\). Combining the results, we get the final answer: \(2x^4 - 6x^3 + 2x^2\).

Step by step solution

01

Identify the monomial and trinomial

First, we need to identify the monomial and trinomial given in the problem. A monomial is a polynomial with only one term, and a trinomial is a polynomial with three terms. For example, let the monomial be 2x^2 and the trinomial be x^2 - 3x + 1.
02

Distribute the monomial

Now, we need to distribute the monomial term across each term in the trinomial. To do this, we will multiply the monomial by each term in the trinomial one by one.
03

Multiply the monomial with the first term of the trinomial

Start by multiplying the monomial with the first term in the trinomial: \(2x^2 * x^2\). Using the rules for exponents, this multiplication can be simplified to \(2x^4\).
04

Multiply the monomial with the second term of the trinomial

Next, multiply the monomial with the second term in the trinomial: \(2x^2 * (-3x)\). This multiplication can be simplified to \(-6x^3\).
05

Multiply the monomial with the third term of the trinomial

Finally, multiply the monomial with the third term in the trinomial: \(2x^2 * 1\). This multiplication can be simplified to \(2x^2\).
06

Combine the results

Now that we have distributed the monomial across each term in the trinomial, we need to combine the results: \(2x^4 - 6x^3 + 2x^2\). As no further simplification is possible, this is the final answer. So, when multiplying a monomial by a trinomial, the result is \(2x^4 - 6x^3 + 2x^2\).

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Key Concepts

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

Monomial
A monomial is a single term in a polynomial and can consist of a constant, a variable, or a product of constants and variables. The defining feature is its single-term nature.
For example, in the expression \(3x^2\), you have a monomial as it consists of a coefficient (3) and a variable raised to a power (\(x^2\)).
It's important to remember:
  • It doesn't have any additions or subtractions within it.
  • It can still have coefficients and variables multiplied together.
In contrast, terms with two or more parts separated by addition or subtraction are not monomials.
To multiply a monomial by another polynomial, you must apply it to each term of the other polynomial individually.
This concept easily leads into the next section, where we tackle trinomials.
Trinomial
A trinomial is a type of polynomial that contains exactly three terms, each of which is separated by either addition or subtraction.
An example of a trinomial is \(x^2 - 3x + 1\).
Here鈥檚 what makes a trinomial unique:
  • It consists of three distinct terms.
  • Each term can be a constant or involve a variable (or variables) raised to some power.
Understanding the structure of a trinomial is crucial when multiplying another polynomial by it.
When a monomial is multiplied by a trinomial, each term in the trinomial receives the monomial as a factor, changing the powers and coefficients of the original terms.
Distributive Property
The distributive property is a fundamental principle in algebra that facilitates operations across terms within parentheses.
It states that \(a(b + c) = ab + ac\).
Applying it to our problem:
  • Each term within the trinomial gets multiplied by the monomial.
  • Each operation is performed separately before combining results.
In practice, this means taking the monomial and multiplying it by each term inside the trinomial one-by-one.
This breaks down a potentially complex multiplication into manageable steps that are easier to simplify.
It's especially useful in understanding how polynomials interact during multiplication, leading to refined results and deeper insight into algebraic expressions.

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