Problem 27 Find each product. $$ -4 r^{... [FREE SOLUTION]

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Chapter 4: Problem 27

Find each product. $$ -4 r^{3}\left(-7 r^{2}+8 r-9\right) $$

Short Answer

Expert verified

The product is \[28r^5 - 32r^4 + 36r^3\]

Step by step solution

- Distribute $-4r^3$

Distribute $-4r^3$ to each term inside the parentheses $(-7r^2 + 8r - 9)$.

- Multiply $-4r^3$ with $-7r^2$

Multiply $-4r^3 $ with $-7r^2$. This gives \[-4r^3 \times (-7r^2) = 28r^5\]

- Multiply $-4r^3$ with $+8r$

Multiply $-4r^3$ with $+8r$. This gives \[-4r^3 \times 8r = -32r^4\]

- Multiply $-4r^3$ with $-9$

Multiply $-4r^3$ with $-9$. This gives \[-4r^3 \times (-9) = 36r^3\]

- Combine all terms

Combine all the resulting terms to get the final product: \[28r^5 - 32r^4 + 36r^3\]

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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 crucial concept in algebra. It helps to simplify expressions and solve equations efficiently. This property states that if you multiply a number by a sum, it's the same as multiplying that number by each addend separately and then adding the products. In symbolic form, it's expressed as:
\[ a(b + c) = ab + ac \]
In our exercise, we're using the distributive property to multiply: \[ -4r^3(-7r^2 + 8r - 9) \]
Here, $ -4r^3 $ is distributed to each term inside the parentheses. This gives us:

\[ -4r^3 \times (-7r^2) \]
\[ -4r^3 \times 8r \]
\[ -4r^3 \times (-9) \]

Understanding this property makes working with polynomials much more manageable.

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers. When multiplying terms with the same base, you add their exponents. The general rule is:
\[ a^m \times a^n = a^{m+n} \]
In our problem, we apply this rule when distributing $ -4r^3 $:

For \[ -4r^3 \times (-7r^2) \], we multiply the coefficients $ -4 \times -7 = 28 $
and add the exponents: \[ r^{3+2} = r^5 \]. This results in \[ 28r^5 \].
For \[ -4r^3 \times 8r \], we multiply the coefficients $ -4 \times 8 = -32 $
and add the exponents: \[ r^{3+1} = r^4 \]. This results in \[ -32r^4 \].
For \[ -4r^3 \times (-9) \], we multiply $ -4 \times -9 = 36 $
and keep the exponent on $ r^3 $ as there's no r in the other term. This results in \[ 36r^3 \].

Mastering these exponent rules is key to dealing with polynomial multiplication efficiently.

Combining Like Terms

Combining like terms is the process of simplifying an expression by summing coefficients of terms with the same variable and exponent. This helps to streamline the final expression. In our example, after distribution, the terms are already in their simplest form:

\[ 28r^5 \]
\[ -32r^4 \]
\[ 36r^3 \]

Since there are no other terms to combine, we simply write them together to get the final product: \[ 28r^5 - 32r^4 + 36r^3 \]
By carefully identifying and combining like terms, we simplify complex polynomial expressions into a more manageable form.

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91影视

Find each product. $$ -4 r^{3}\left(-7 r^{2}+8 r-9\right) $$

Short Answer

Step by step solution

- Distribute \(-4r^3\)

- Multiply \(-4r^3\) with \(-7r^2\)

- Multiply \(-4r^3\) with \(+8r\)

- Multiply \(-4r^3\) with \(-9\)

- Combine all terms

Key Concepts

Distributive Property

Exponent Rules

Combining Like Terms

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