Problem 26 Find each product. $$ 4 z^{3... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 5: Problem 26

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

Short Answer

Expert verified

32z^5 + 20z^4y - 12z^3y^2

Step by step solution

- Distribute 4z^3 to Each Term

Multiply each term inside the parentheses by the term outside. Start by distributing the term $4z^3$ to each term inside the parentheses: $8z^2, 5zy, -3y^2$.

- Multiply $4z^3$ by $8z^2$

Multiply $4z^3$ by $8z^2$: $4z^3 \times 8z^2 = 32z^{5}$.

- Multiply $4z^3$ by $5zy$

Multiply $4z^3$ by $5zy$: $4z^3 \times 5zy = 20z^4y$.

- Multiply $4z^3$ by $-3y^2$

Multiply $4z^3$ by $-3y^2$: $4z^3 \times -3y^2 = -12z^3y^2$.

- Combine All Terms

Combine all the multiplied terms to get the final product: $32z^5 + 20z^4y - 12z^3y^2$.

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Key Concepts

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

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a set of parentheses. This property is essential when dealing with polynomial multiplication. In the given exercise, we need to distribute $4z^3$ to each term inside the parentheses: $8z^2 + 5zy - 3y^2$.

Let鈥檚 break this down:

We multiply $4z^3$ by $8z^2$ to get $32z^5$.
Next, we multiply $4z^3$ by $5zy$ to get $20z^4y$.
Finally, we multiply $4z^3$ by$-3y^2$ to get $-12z^3y^2$.

By distributing correctly, we ensure that each term within the parentheses is accounted for. Combining these products gives us the final result: $32z^5 + 20z^4y - 12z^3y^2$.

Exponent Rules

Understanding exponent rules is crucial when performing polynomial multiplication. The two main rules to remember are:

When multiplying like bases, add the exponents: If you have $a^m \times a^n$, the result is $a^{m+n}$.
When raising a power to a power, multiply the exponents: If you have $(a^m)^n$, the result is $a^{m \times n}$.

In our exercise, as we multiply terms, we need to apply the first rule. For instance:

$4z^3 \times 8z^2 = 32z^{3+2} = 32z^5$.
$4z^3 \times 5zy = 20z^{3+1}y = 20z^4y$.
$4z^3 \times -3y^2 = -12z^3y^2$ (since there is no z in the second term, we keep the exponents as they are).

These simple rules help simplify expressions quickly and accurately.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. It represents a specific value.

In the exercise, our initial expression is $4z^3(8z^2 + 5zy - 3y^2)$.

To solve it:

We distribute $4z^3$, as explained before, to each term within the parentheses. Each multiplication follows the exponent rules.
This results in each term being simplified to $32z^5, 20z^4y,$ and $-12z^3y^2$.
Combining these gives us the final algebraic expression: $32z^5 + 20z^4y - 12z^3y^2$.

Algebraic expressions are used across mathematics to generalize rules and formulas, paving the way for more complex problem-solving.

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91影视

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

Short Answer

Step by step solution

- Distribute 4z^3 to Each Term

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

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

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

- Combine All Terms

Key Concepts

Distributive Property

Exponent Rules

Algebraic Expressions

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