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Chapter 4: Problem 30

Find each product. $$ 4 z^{3}\left(8 z^{2}+5 z y-3 y^{2}\right) $$

Short Answer

Expert verified
Combine to get: \(32z^5 + 20z^4y - 12z^3y^2\).

Step by step solution

01

- Distribute each term

Multiply the term outside the parentheses, \(4z^3\), by each term inside the parentheses. The expression inside the parentheses is \(8z^2 + 5zy - 3y^2\).
02

- Multiply \(4z^3\) by \(8z^2\)

\(4z^3 \times 8z^2 = 32z^5\). Write this term down.
03

- Multiply \(4z^3\) by \(5zy\)

\(4z^3 \times 5zy = 20z^4y\). Write this term down.
04

- Multiply \(4z^3\) by \(-3y^2\)

\(4z^3 \times (-3y^2) = -12z^3y^2\). Write this term down.
05

- Combine all terms

Now combine all the terms you obtained from the distribution: \(32z^5 + 20z^4y - 12z^3y^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operators (like addition and multiplication) without an equality sign. For example, in the expression we were multiplying, \(4z^3(8z^2 + 5zy - 3y^2)\), both \(4z^3\) and \((8z^2 + 5zy - 3y^2)\) are algebraic expressions. They include:
  • Constants: Numbers like 4, 8, 5, -3.
  • Variables: Letters that represent numbers, like \(z\) and \(y\).
  • Exponential terms: Variables raised to some power, like \(z^3\) and \(y^2\).
An algebraic expression can represent complex relationships and functions in math by combining these elements in different ways.
Distributive Property
The distributive property allows us to simplify expressions by distributing multiplication over addition or subtraction. Essentially, when you have \(a(b + c)\), it can be rewritten as \(ab + ac\). In our exercise, we use the distributive property to multiply \(4z^3\) by each term inside the parentheses \((8z^2 + 5zy - 3y^2)\):
  • First, \(4z^3 \times 8z^2 = 32z^5\)
  • Next, \(4z^3 \times 5zy = 20z^4y\)
  • Finally, \(4z^3 \times (-3y^2) = -12z^3y^2\)
After distributing and simplifying each multiplication, the resultant terms are combined to form the final expression: \(32z^5 + 20z^4y - 12z^3y^2\).
Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication, and involving non-negative integer exponents. In the given exercise, the final result \(32z^5 + 20z^4y - 12z^3y^2\) is a polynomial. Here are a few key characteristics of polynomials:
  • Each term consists of a coefficient (number) and a variable raised to an exponent.
  • They can have one or more terms. For instance, \(32z^5\) is a single-term polynomial (monomial), and \(32z^5 + 20z^4y - 12z^3y^2\) is a three-term polynomial (trinomial).
  • Polynomials can be used to represent a wide range of mathematical relationships and functions.
Understanding polynomials is fundamental to mastering more advanced topics in algebra and calculus.

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Most popular questions from this chapter

Add or subtract as indicated. \((8 a b+2 a-3 b)-(6 a b-2 a+3 b)\)

$$ \begin{aligned} &y=-x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 76\. \(\frac{0.000000081(0.000036)}{0.00000048}\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (3 r-2 s)^{4} $$

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. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$
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