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Chapter 5: Problem 68

Find each product. In each case, neither factor is a monomial. $$(x+2)\left(x^{2}+x+5\right)$$

Short Answer

Expert verified
The product of \((x+2)\) and \(\left(x^{2}+x+5\right)\) is: \(x^{3} + 3x^{2} + 7x + 10\)

Step by step solution

01

Distribute the First Term

Distribute \(x\) (the first term of the first polynomial) to each term of the second polynomial: \(x * x^{2}\), \(x * x\), \(x * 5\). This gives us: \(x^{3} + x^{2} + 5x\).
02

Distribute the Second Term

Distribute \(2\) (the second term of the first polynomial) to each term of the second polynomial: \(2 * x^{2}\), \(2 * x\), \(2 * 5\). This gives us: \(2x^{2} + 2x + 10\).
03

Combine Like Terms

Combine the results to get the final polynomial: \(x^{3} + x^{2} + 5x + 2x^{2} + 2x + 10\). The result is: \(x^{3} + 3x^{2} + 7x + 10\).

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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 fundamental tool in algebra used to multiply a single term by each term within a polynomial. When faced with expressions like
  • \((x+2)(x^2+x+5)\),
we apply the distributive property step by step. First, take the first term of the first expression:
  • Distribute \(x\) to every term of the second polynomial \(x^2 + x + 5\):
    • \(x imes x^2 = x^3\)
    • \(x imes x = x^2\)
    • \(x imes 5 = 5x\)
  • Then, use the second term \(2\) in the same way:
    • \(2 imes x^2 = 2x^2\)
    • \(2 imes x = 2x\)
    • \(2 imes 5 = 10\)
This process ensures that each part of one polynomial interacts with every part of the other, helping us build the complete expression.
Combining Like Terms
After using the distributive property, you'll face multiple terms. Some of these terms may be similar, meaning they contain the same variable raised to the same power. In algebra, we sum these like terms together:

  • Begin with the expression from our example:
    • \(x^3 + x^2 + 5x + 2x^2 + 2x + 10\)
  • Identify the like terms by their exponents:
    • \(x^2\) terms: \(x^2\) and \(2x^2\)
    • \(x\) terms: \(5x\) and \(2x\)
  • Combine like terms:
    • \(x^2 + 2x^2 = 3x^2\)
    • \(5x + 2x = 7x\)
  • Include terms with no like partners:
    • \(x^3\)
    • Constant \(10\)
By aligning like terms, the complex expression simplifies into \(x^3 + 3x^2 + 7x + 10\). Each term's properties are preserved, streamlining your solution.
Polynomial Expressions
Polynomial expressions are sums of terms consisting of variables raised to non-negative integer powers and multiplied by coefficients. They're used in numerous real-world problems and extend our understanding of algebra significantly. Here's what makes them unique:
  • Variables: These could be any symbol like \(x\), and they can appear in differing degrees (powers) throughout the polynomial.
  • Terms: A polynomial is composed of terms such as \(2x^2\) or \(-7x\), which include a variable raised to a power and a coefficient.
  • Degrees: The degree of a polynomial is determined by the highest exponent. For instance, in \(x^3 + 3x^2 + 7x + 10\), the degree is three, given the term \(x^3\).
Understanding polynomial expressions means recognizing these structures and working through operations like addition, subtraction, and multiplication. By mastering these, you鈥檒l tackle broader problems with confidence.

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Most popular questions from this chapter

What polynomial, when divided by \(3 x^{2},\) yields the trinomial \(-6 x^{6}-9 x^{4}+12 x^{2}\) as a quotient?

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