Problem 58 Find each product. $$ -5 r(r... [FREE SOLUTION]

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Chapter 4: Problem 58

Find each product. $$ -5 r(r+1)^{3} $$

Short Answer

Expert verified

The product is $-5r^4 - 15r^3 - 15r^2 - 5r$.

Step by step solution

Understand the Expression

The given expression is \text{$-5r(r+1)^3$}. It involves multiplying $-5r$ with $(r+1)^3$.

Expand $(r+1)^3$

First expand $(r+1)^3$: $(r+1)^3$ is the same as $(r+1)(r+1)(r+1)$. Use the binomial theorem or multiply it step-by-step. When expanded, $(r+1)^3 = r^3 + 3r^2 + 3r + 1$.

Distribute $-5r$ to Each Term

The expression $-5r(r^3 + 3r^2 + 3r + 1)$ will now be simplified by distributing $-5r$ to each term inside the parenthesis:$-5r \times r^3 = -5r^4$$-5r \times 3r^2 = -15r^3$$-5r \times 3r = -15r^2$$-5r \times 1 = -5r$.

Combine All Parts

Combine all the terms from the previous step: $-5r^4$, $-15r^3$, $-15r^2$, and $-5r$. The final product is $-5r^4 - 15r^3 - 15r^2 - 5r$.

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Key Concepts

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

Polynomial Expansion

Polynomial expansion involves breaking down expressions like $(r+1)^3$ into a sum of terms. This is especially useful for simplifying and solving problems.

For example, in the exercise $-5r(r+1)^3$, we need to expand $ (r+1)^3$. By multiplying $ (r+1)$ three times, we get:

$(r+1)(r+1)(r+1) = (r^3 + 3r^2 + 3r + 1)$

Always remember to carefully multiply each term inside the parentheses to make sure you cover all the needed combinations for accurate expansion.

Binomial Theorem

The Binomial Theorem offers a quick way to expand expressions of the form $ (a+b)^n$ without manually multiplying each term.

The theorem states: \[ (a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n \]

In our exercise: Expand $ (r+1)^3$ by using the binomial theorem. For $ (r+1)^3 $ it鈥檚 simplified to:
\ r^3 + 3r^2 + 3r + 1\

Applying the binomial theorem makes sure you do not miss any internal term combinations, ensuring accuracy in your polynomial expansion.

Distribution Property

The distribution property, or distributive property, lets you multiply a single term with each term inside a set of parentheses.

For our example $-5r(r+1)^3$:
After expanding $(r+1)^3$ to $r^3 + 3r^2 + 3r + 1$, we distribute $-5r$ to each term.

The steps would be:

$-5r \times r^3 = -5r^4$
$-5r \times 3r^2 = -15r^3$
$-5r \times 3r = -15r^2$
$-5r \times 1 = -5r$

Combining all parts, we get the final product: $-5r^4 - 15r^3 - 15r^2 - 5r$. This approach ensures each term is correctly computed and simplifies the problem effectively.

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91影视

Find each product. $$ -5 r(r+1)^{3} $$

Short Answer

Step by step solution

Understand the Expression

Expand \((r+1)^3\)

Distribute \(-5r\) to Each Term

Combine All Parts

Key Concepts

Polynomial Expansion

Binomial Theorem

Distribution Property

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