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Chapter 4: Problem 23

Find each product. $$ -8 z\left(2 z+3 z^{2}+3 z^{3}\right) $$

Short Answer

Expert verified
-16z^2 - 24z^3 - 24z^4

Step by step solution

01

- Distribute the term

Distribute \(-8z\) to each term inside the parenthesis. This results in \(-8z \times 2z\), \(-8z \times 3z^2\), and \(-8z \times 3z^3\).
02

- Calculate each product

Compute each multiplication: \(-8z \times 2z = -16z^2\), \(-8z \times 3z^2 = -24z^3\), and \(-8z \times 3z^3 = -24z^4\).
03

- Combine the results

Combine the results from Step 2 to create the final expression: \(-16z^2 - 24z^3 - 24z^4\).

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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 rule in algebra that lets you simplify expressions. Think of it as a way to 'distribute' multiplication over addition or subtraction inside parentheses. In simpler terms, if you have a term outside the parentheses, you multiply it with each term inside the parentheses.
For example, in the expression \(a(b + c)\), you distribute 'a' to both 'b' and 'c'.
  • \(a \times b + a \times c\)
In our given problem, \(-8z(2z + 3z^2 + 3z^3)\), we distribute \(-8z\) to each term inside the parentheses:
  • \(-8z \times 2z\)
  • \(-8z \times 3z^2\)
  • \(-8z \times 3z^3\)
This makes it easier to deal with the expression by breaking it into smaller, more manageable parts.
polynomial multiplication
Polynomial multiplication is the process of multiplying two or more polynomials together. Each term in one polynomial is multiplied by each term in the other polynomial. Let's explore how it applies to our example:
Once we distribute \(-8z\) across all terms inside the parentheses, we compute:
  • \(-8z \times 2z = -16z^2\)
  • \(-8z \times 3z^2 = -24z^3\)
  • \(-8z \times 3z^3 = -24z^4\)
Notice that each term on the right side is a product of the terms being multiplied. The left side shows the original terms from the problem. This step-by-step approach clarifies how distributing and multiplying terms helps in breaking down complex problems into simpler ones.
combining like terms
Combining like terms is an important step that involves simplifying the expression by merging terms with the same variables and powers.
After performing the polynomial multiplication in our example, we're left with: \(-16z^2 - 24z^3 - 24z^4\). Now we need to check for any like terms to combine. Like terms have the same variable raised to the same power in this case:
  • If there were any \(z^2\) terms, we would add or subtract their coefficients.
  • The same goes for \(z^3\) and \(z^4\) terms.
In our expression, none of the terms share the same variable and power, so they remain as is. Combining like terms simplifies expressions, but it only works if those terms have identical variables raised to the same power. This ensures you end up with the simplest form of your answer.

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Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m+6)^{2} $$

Perform each indicated operation. \(\left(x^{2}+x\right)-\left(3 x^{2}+2 x-1\right)\)

Light travels at a speed of \(1.86 \times 10^{5} \mathrm{mi}\) per sec. When Venus is \(6.68 \times 10^{7} \mathrm{mi}\) from the sun, how long (in seconds) does it take light to travel from the sun to Venus? (Data from The World Almanac and Book of Facts.)

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 73\. \(\frac{650,000,000(0.0000032)}{0.00002}\)
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