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Chapter 5: Problem 69

Multiply using the method of your choice. $$3 x^{2}\left(4 x^{2}+x+9\right)$$

Short Answer

Expert verified
So, the result of multiplying \(3x^{2}\) and \(4x^{2}+x+9\) using the distributive property is \(12x^{4} + 3x^{3} + 27x^{2}\)

Step by step solution

01

Apply the distributive property

We distribute \(3x^{2}\) across each term in the parenthesis so that it multiplies each term. \(3x^{2} \times 4x^{2}\), \(3x^{2} \times x\), and \(3x^{2} \times 9\). This operation gives us \(12x^{4} + 3x^{3} + 27x^{2}\)
02

Check the result

Ensure that the multiplication and distribution operations have been carried out correctly. That leaves us with the final expression: \(12x^{4} + 3x^{3} + 27x^{2}\)

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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 key component in algebra that allows you to break down complex expressions into simpler parts. When you apply the distributive property, you multiply a single term by each of the terms inside a set of parentheses. This property is essential when you want to simplify expressions or solve equations.
In our exercise, we used the distributive property to multiply each term within the parentheses by the term outside, which was \(3x^2\).
  • First, multiply \(3x^2\) by \(4x^2\) to get \(12x^4\).
  • Next, multiply \(3x^2\) by \(x\) to get \(3x^3\).
  • Lastly, multiply \(3x^2\) by \(9\) to get \(27x^2\).
This process shows how the distributive property helps us expand and simplify polynomial expressions by distributing each term individually.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are the basic building blocks for solving equations and help represent real-world problems in mathematical terms.
An expression like \(3x^2(4x^2 + x + 9)\) consists of several parts:
  • Coefficients: The numbers that multiply the variables in the expression. For example, \(3\) is the coefficient of \(x^2\) in our task.
  • Variables: The symbols used to represent unknown quantities. In our example, \(x\) is the variable.
  • Constants: Numbers without variables. Here, \(9\) is a constant term.
Unfortunately, these expressions do not include an equals sign, meaning they don't form equations but can be manipulated using various algebraic properties to achieve a desired form or solve a larger problem.
Mathematical Operations
Mathematical operations are procedures that take one or more numbers, called operands, and produce a new number, the result. In the context of the multiplication of polynomials, we primarily utilize multiplication and addition.
In our exercise, the operation of multiplication was used repeatedly to distribute \(3x^2\) across the polynomial expression in the parentheses.
  • Multiplication: Each term in the expression was multiplied, increasing the exponent of the variable where applicable.
  • Addition: After all terms were individually multiplied, the results were added together to form a single polynomial expression.
These operations allow us to expand and combine expressions to simplify or solve problems that involve algebraic expressions. Mastery of these foundational operations is imperative for successfully handling more complex algebraic and mathematical tasks.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{\left(x^{2}\right)^{3}}{5^{3}}$$

Use the order of operations to evaluate $$x^{3} y+2 x y^{2}+5 x-2$$ for \(x=-2\) and \(y=3\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(3 x^{2}+2\right)\left(3 x^{2}-2\right)=9 x^{2}-4$$

What is the degree of a polynomial? Provide an example with your explanation.

What is a binomial? Give an example with your explanation.
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