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Chapter 1: Problem 88

$$\text { Multiply: }\left(3 x^{2}-x+2\right)(2 x-3)[\text { A. } 2]$$

Short Answer

Expert verified
The result of the multiplication is \(6x^3 - x^2 + 4x - 6\).

Step by step solution

01

Multiply Each Term from the Binomial with Each Term from the Trinomial

First, multiply each term from the binomial by each term of the trinomial accordingly. You get: \(2x * 3x^2=6x^3, -3 * 3x^2=-9x^2, 2x * -x=-2x^2, -3 * -x=3x, 2x * 2=4x, -3 * 2=-6\)
02

Combine Like Terms

Next, combine like terms from the products calculated in the first step: \(6x^3-x^2+4x-6\)
03

Final Simplification

As a last step, you will simplify the result to its lowest term. But in this case, it is already simplified and there is nothing to reduce further, so Step 2's result is the final answer

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Key Concepts

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

Binomial
In algebra, a binomial is an expression containing two terms separated by a plus (+) or minus (-) sign. It is one of the fundamental building blocks when dealing with polynomials. In our exercise, the binomial is \(2x - 3\). Each of these terms can be a variable (like \(x\)) with coefficients (like 2) or a constant, such as -3. Understanding a binomial is crucial for performing operations like addition, subtraction, multiplication, and division of polynomials.
Knowing how to manipulate binomials helps in recognizing and applying the distributive property efficiently. This is essential when multiplying a binomial with another polynomial term, such as a trinomial.
Trinomial
A trinomial is an algebraic expression made up of three terms. In our context, \(3x^2 - x + 2\) is the trinomial. Similar to a binomial, these could be a combination of variables and constants. Each term in a trinomial has its distinct part and can be a constant or include variable components.
When performing operations with trinomials, it's helpful to understand each term's role and how it fits into the whole expression. This helps you avoid mistakes when applying the distributive property to multiply them with another polynomial, as with a binomial.
Combining Like Terms
In algebra, combining like terms is a method used to simplify expressions by adding or subtracting coefficients of terms with the same variables and exponents. It's an essential step toward simplifying expressions.
When multiplying a binomial by a trinomial, you often end up with several terms that share the same variables and exponents. For example, in the solution, we have terms like \(-9x^2\) and \(-2x^2\) resulting in \(-11x^2\) when combined.
  • Combine terms with the same powers of variables.
  • Always ensure accuracy in arithmetic operations.
This step is crucial for arriving at the correct simplified expression.
Simplification
Simplification in algebra is the process of reducing an expression to its most concise and clear form. After operations like multiplication and combining like terms, the expression should be checked for any further simplifications.
In our example, after following all necessary algebraic steps, the expression \(6x^3 - 11x^2 + 4x - 6\) is already in its simplest form. It's important to ensure no additional factors can be extracted and to confirm no further combinations of terms are possible.
Simplifying ensures the expression is easy to work with in case more operations are needed or while checking for equivalence with other expressions.

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Most popular questions from this chapter

Use interval notation to express the solution set of each inequality. $$|4-5 x| \geq 24$$

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Use the properties of inequalities to solve each inequality. Write answers using interval notation. $$-3(x+2) \leq 5 x+7$$
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