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Chapter 6: Problem 98

Multiply the following. See Section 5.4 \((z-2)\left(z^{2}+2 z+4\right)\)

Short Answer

Expert verified
The product is \(z^3 - 8\).

Step by step solution

01

Distribute the First Term

To multiply \(z-2\) and \(z^2+2z+4\), we'll start by distributing the first term of the binomial, which is \(z\).Multiply \(z\) by each term in the trinomial:- \(z imes z^2 = z^3\)- \(z imes 2z = 2z^2\)- \(z imes 4 = 4z\)This gives us \(z^3 + 2z^2 + 4z\).
02

Distribute the Second Term

Next, distribute the second term of the binomial, which is \(-2\).Multiply \(-2\) by each term in the trinomial:- \(-2 imes z^2 = -2z^2\)- \(-2 imes 2z = -4z\)- \(-2 imes 4 = -8\)This gives us \(-2z^2 - 4z - 8\).
03

Combine Like Terms

Now combine the results from Steps 1 and 2:\(z^3 + 2z^2 + 4z\) and \(-2z^2 - 4z - 8\).Group and add the like terms:- \(z^3\) is by itself so it remains \(z^3\).- Combine \(2z^2\) and \(-2z^2\) to get \(0\).- Combine \(4z\) and \(-4z\) to get \(0\).- The constant term is \(-8\).Thus, the final expression is \(z^3 - 8\).

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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 an essential tool in algebra. It allows you to multiply a single term by each term in a polynomial expression.
In our example,
  • Begin with the expression \((z-2)(z^2+2z+4)\).
  • The first term of the binomial, \(z\), is distributed across every term in the trinomial:
    • \(z \times z^2 = z^3\)
    • \(z \times 2z = 2z^2\)
    • \(z \times 4 = 4z\)
    This results in \(z^3 + 2z^2 + 4z\).
  • Next, distribute the second term, \(-2\):
    • \(-2 \times z^2 = -2z^2\)
    • \(-2 \times 2z = -4z\)
    • \(-2 \times 4 = -8\)
    Giving us \(-2z^2 - 4z - 8\).
This step ensures that every term from the binomial interacts with every term in the trinomial.
Combining Like Terms
Combining like terms is crucial to simplifying an expression. After using the distributive property, you might have terms that can be combined.
Like terms are those that have the same variable raised to the same power.
  • In the expression \(z^3 + 2z^2 + 4z\) and \(-2z^2 - 4z - 8\):
    • \(z^3\) has no like term to combine with, so it remains as \(z^3\).
    • The terms \(2z^2\) and \(-2z^2\) cancel each other out, as they add to zero.
    • Similarly, \(4z\) and \(-4z\) also cancel each other out, again, summing up to zero.
    • The constant \(-8\) stands alone, so it remains unchanged.
This results in a simplified expression of \(z^3 - 8\), removing any unnecessary terms.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. Understanding these expressions is important to solving algebra problems.
Our goal when working with expressions is often to simplify or solve them.
  • In our example, we started with a more complicated product of two expressions: \((z-2)(z^2 + 2z + 4)\).
  • The expression was simplified using polynomial multiplication and resulted in \(z^3 - 8\).
  • This means breaking down the expression and following algebraic rules to get it to its simplest form.
Algebraic expressions form the backbone of algebra. Understanding how to manipulate them is key to higher-level math concepts.

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Most popular questions from this chapter

Factor each binomial completely. \(36 x^{2} y-25 y\)

Factor each binomial completely. \(18 r^{2}-8\)

Factor each trinomial completely. See Examples I through II and Section 6.2. \(1+16 x^{2}+x^{4}\)

Factor each trinomial completely. See Examples I through II and Section 6.2. \(m^{3}+18 m^{2}+81 m\)

Factor each trinomial completely. See Examples 1 through 11 and Section 6.2. \(-27 t+7 t^{2}-4\)
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