– 4d (5ď² – 12) + 7 (d + 5)

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Mathematical Expression for Educational Content

The given mathematical expression is:

\[ 2a \left( 7a^2b^2 + ab^2 + 5b \right) - 9b \left( -2a^2b^2 + 2b^2 + 3a \right) \]

This expression involves two main components:

1. The first term inside the parentheses, \(2a (7a^2b^2 + ab^2 + 5b)\), where:
   - \(7a^2b^2\) is a term involving both \(a\) and \(b\) raised to powers of 2.
   - \(ab^2\) is a product of \(a\) and \(b\), with \(b\) raised to the power of 2.
   - \(5b\) is a linear term in \(b\).

2. The second term inside the parentheses, \(-9b (-2a^2b^2 + 2b^2 + 3a)\), where:
   - \(-2a^2b^2\) includes a negative sign and involves both \(a\) and \(b\) raised to powers of 2.
   - \(2b^2\) is a quadratic term in \(b\).
   - \(3a\) is a linear term in \(a\). 

#### Detailed Breakdown:
- **First Parenthesis**: Multiply \(2a\) by each term inside the parenthesis:
  \[ 2a \left( 7a^2b^2 + ab^2 + 5b \right) = (2a)(7a^2b^2) + (2a)(ab^2) + (2a)(5b) = 14a^3b^2 + 2a^2b^2 + 10ab \]
  
- **Second Parenthesis**: Multiply \(-9b\) by each term inside the parenthesis:
  \[ -9b \left( -2a^2b^2 + 2b^2 + 3a \right) = (-9b)(-2a^2b^2) + (-9b)(2b^2) + (-9b)(3a) = 18a^2b^3 -
Transcribed Image Text:### Mathematical Expression for Educational Content The given mathematical expression is: \[ 2a \left( 7a^2b^2 + ab^2 + 5b \right) - 9b \left( -2a^2b^2 + 2b^2 + 3a \right) \] This expression involves two main components: 1. The first term inside the parentheses, \(2a (7a^2b^2 + ab^2 + 5b)\), where: - \(7a^2b^2\) is a term involving both \(a\) and \(b\) raised to powers of 2. - \(ab^2\) is a product of \(a\) and \(b\), with \(b\) raised to the power of 2. - \(5b\) is a linear term in \(b\). 2. The second term inside the parentheses, \(-9b (-2a^2b^2 + 2b^2 + 3a)\), where: - \(-2a^2b^2\) includes a negative sign and involves both \(a\) and \(b\) raised to powers of 2. - \(2b^2\) is a quadratic term in \(b\). - \(3a\) is a linear term in \(a\). #### Detailed Breakdown: - **First Parenthesis**: Multiply \(2a\) by each term inside the parenthesis: \[ 2a \left( 7a^2b^2 + ab^2 + 5b \right) = (2a)(7a^2b^2) + (2a)(ab^2) + (2a)(5b) = 14a^3b^2 + 2a^2b^2 + 10ab \] - **Second Parenthesis**: Multiply \(-9b\) by each term inside the parenthesis: \[ -9b \left( -2a^2b^2 + 2b^2 + 3a \right) = (-9b)(-2a^2b^2) + (-9b)(2b^2) + (-9b)(3a) = 18a^2b^3 -
The given mathematical expression is:

\[ -4d (5d^2 - 12) + 7 (d + 5) \]

This expression involves the variable \( d \) and includes terms with different degrees of \( d \). It demonstrates polynomial multiplication and addition. The operations need to be carried out as follows:

1. First, distribute \(-4d\) through the polynomial \(5d^2 - 12\).
2. Second, distribute \(7\) through the polynomial \(d + 5\).

Breaking it down step-by-step:

\[ -4d \cdot 5d^2 - (-4d) \cdot 12 + 7 \cdot d + 7 \cdot 5 \]

This leads to:

\[ -20d^3 + 48d + 7d + 35 \]

Combining like terms results in:

\[ -20d^3 + 55d + 35 \]

This is the simplified form of the given expression. Understanding these steps helps in mastering polynomial operations, which are fundamental in algebra.
Transcribed Image Text:The given mathematical expression is: \[ -4d (5d^2 - 12) + 7 (d + 5) \] This expression involves the variable \( d \) and includes terms with different degrees of \( d \). It demonstrates polynomial multiplication and addition. The operations need to be carried out as follows: 1. First, distribute \(-4d\) through the polynomial \(5d^2 - 12\). 2. Second, distribute \(7\) through the polynomial \(d + 5\). Breaking it down step-by-step: \[ -4d \cdot 5d^2 - (-4d) \cdot 12 + 7 \cdot d + 7 \cdot 5 \] This leads to: \[ -20d^3 + 48d + 7d + 35 \] Combining like terms results in: \[ -20d^3 + 55d + 35 \] This is the simplified form of the given expression. Understanding these steps helps in mastering polynomial operations, which are fundamental in algebra.
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