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Chapter 5: Problem 198

Multiply the binomials. Use any method. $$ \left(x^{2}+3\right)(x+2) $$

Short Answer

Expert verified
The product is \( x^{3} + 2x^{2} + 3x + 6 \).

Step by step solution

01

- Distribute each term in the first binomial

Multiply each term in the binomial \((x^{2}+3) \) with each term in the binomial \((x+2) \): \[((x^{2})(x) + (x^{2})(2) + (3)(x) + (3)(2)) \]
02

- Perform the multiplications

Calculate the multiplications for each pair of terms: \[(x^{2})(x) = x^{3}, (x^{2})(2) = 2x^{2}, (3)(x) = 3x, (3)(2) = 6. \]So, the expression becomes: \[x^{3} + 2x^{2} + 3x + 6 \]

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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 helps us to multiply a single term across terms inside a set of parentheses. We use this property frequently, especially when dealing with polynomials.

For example, if you have two binomials such as \((x^2 + 3)(x + 2)\), you need to distribute each term in the first binomial to every term in the second binomial. This essentially means taking each term from \((x^2 + 3)\) and multiplying it by every term in \((x + 2)\).

So, we get:
\[ (x^2)(x) + (x^2)(2) + (3)(x) + (3)(2) \]

This breaks it down into manageable pieces which we can then further simplify by performing the multiplications.
polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, organized in terms of their degrees in descending order.

For example, in our exercise, we started with binomials \((x^2 + 3)\) and \((x + 2)\).
When we multiply these two binomials, we apply the distributive property to obtain the expanded polynomial: \[ x^3 + 2x^2 + 3x + 6 \]

Each term in this polynomial is an individual monomial, where:
  • \[x^3 \] is a term with a degree of 3
  • \[ 2x^2 \] is a term with a degree of 2
  • \[3x \] is a term with a degree of 1
  • \[6 \] is the constant term


This polynomial is now in its simplest form, as there are no like terms to combine.
algebraic expressions
Algebraic expressions involve numbers, variables, and arithmetic operators.

For instance, the given exercise involves the expression \((x^2 + 3)(x + 2) \), which contains:
  • Terms with variables (e.g., \[x^2 \])
  • Arithmetic operations like addition and multiplication
  • A constant term (e.g., \[3 \])


To simplify these expressions, we often use properties such as the distributive property, which enables us to break down and simplify more complex expressions into simpler, more manageable pieces.
intermediate algebra
Intermediate algebra deals with more advanced algebraic concepts and techniques beyond basic algebra. it often involves the manipulation and simplification of polynomials, rational expressions, and radical expressions.

In our exercise, multiplying the binomials \((x^2 + 3) \) and \((x + 2) \) and simplifying the resulting polynomial exemplifies an intermediate algebra topic.

As the complexity increases, it becomes necessary to follow precise steps methodically:
  • Distribute each term
  • Perform the multiplications
  • Combine like terms, if any


Mastery of these techniques prepares students for advanced algebra topics, ensuring that they understand how to manipulate and simplify complex algebraic expressions effectively.

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Most popular questions from this chapter

Multiply the monomials. (a) \(\left(-10 x^{5}\right)\left(-3 x^{3}\right)\) (b) \(\left(\frac{5}{8} x^{3} y\right)\left(24 x^{5} y\right)\)

Find each product. $$ -5 m\left(m^{2}+3 m-18\right) $$

Multiply the binomials. Use any method. $$ (3 r-8)(11 r+1) $$

Find each product. $$ (4 a+10)^{2} $$

Find each product. $$ (m-15)^{2} $$
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