Problem 25 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. \(7\left(a^{3} b^{3}+6 a^{3} b^{2}+3\right)\) \(-9\left(6 a^{3} b^{3}+9 a^{3} b^{2}-12 a^{2} b+8\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(-47a^{3}b^{3} - 39a^{3}b^{2} + 108a^{2}b - 51\).

Step 1: Distribute the given values to each term inside the parentheses

To do this, we will multiply each term inside the parentheses by the value outside. For the first parentheses, distribute the \(7\): \(7(a^{3}b^{3} + 6a^{3}b^{2} + 3) = 7a^{3}b^{3} + 42a^{3}b^{2} + 21\) For the second parentheses, distribute the \(-9\): \(-9(6a^{3}b^{3} + 9a^{3}b^{2} - 12a^{2}b + 8) = -54a^{3}b^{3} - 81a^{3}b^{2} + 108a^{2}b - 72\)

Step 2: Combine like terms

Now that we have distributed the values, we can combine the like terms from both expressions: \((7a^{3}b^{3} + 42a^{3}b^{2} + 21) - (54a^{3}b^{3} + 81a^{3}b^{2} - 108a^{2}b + 72)\) Combine the terms: 7a^{3}b^{3} - 54a^{3}b^{3} = -47a^{3}b^{3} 42a^{3}b^{2} - 81a^{3}b^{2} = -39a^{3}b^{2} 108a^{2}b = 108a^{2}b 21 - 72 = -51

Step 3: Write the simplified expression

Combine all terms to write the simplified expression: \(-47a^{3}b^{3} - 39a^{3}b^{2} + 108a^{2}b - 51\) So the simplified expression is \(\boldsymbol{-47a^{3}b^{3} - 39a^{3}b^{2} + 108a^{2}b - 51}\).

Key Concepts

Distribution

In algebra, distribution is a crucial concept that helps in multiplying a single term across multiple terms within parentheses. This process is guided by the distributive property, which states that multiplying a number (or term) by a sum is the same as multiplying that number by each addend individually and then summing the products.
When applied to polynomials, distribution is key to expanding expressions. For instance, when you have something like \(7(a^3b^3 + 6a^3b^2 + 3)\), you multiply 7 by each term inside the parentheses:

• Multiply 7 by \(a^3b^3\) to get \(7a^3b^3\).
• Multiply 7 by \(6a^3b^2\) to get \(42a^3b^2\).
• Multiply 7 by 3 to get 21.

Similarly, for negative distribution like in \(-9(6a^3b^3 + 9a^3b^2 - 12a^2b + 8)\), make sure to carefully distribute the negative sign alongside the factor:

• Multiply -9 by \(6a^3b^3\), resulting in \(-54a^3b^3\).
• Multiply -9 by \(9a^3b^2\) to get \(-81a^3b^2\).
• Multiply -9 by \(-12a^2b\), which gives \(108a^2b\) due to the negative times a negative rule.
• Multiply -9 by 8 to get -72.

Combining Like Terms

After distributing, the next important step is combining like terms. Like terms are terms that have the same variables raised to the same power. This step helps streamline expressions by aggregating similar terms, making the expression simpler and easier to interpret.
In the expression:\[7a^{3}b^{3} + 42a^{3}b^{2} + 21 - 54a^{3}b^{3} - 81a^{3}b^{2} + 108a^{2}b - 72\]You can identify several like terms:

• \(7a^3b^3\) and \(-54a^3b^3\) are like terms. Combine them to get \(-47a^3b^3\).
• \(42a^3b^2\) and \(-81a^3b^2\) are also like terms. Combine these to yield \(-39a^3b^2\).
• The single term \(108a^2b\) has no like terms, so it remains unchanged.
• The constants 21 and -72 are like terms. Combine them to get -51.

After combining, you are left with a simplified polynomial: \(-47a^{3}b^{3} - 39a^{3}b^{2} + 108a^{2}b - 51\).

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Understanding their structure is essential for performing operations like distribution and combining like terms.
They are typically written in a standard form ordered by the highest power. In the exercise provided, the polynomial consists of terms like \(a^3b^3\) and \(a^2b\), each having specific coefficients:

• The term \(a^3b^3\) has coefficients like 7 and -54 which are combined to get \(-47a^3b^3\).
• The term \(a^3b^2\) becomes \(-39a^3b^2\) after combining 42 and -81.
• The term \(a^2b\) stands alone with a coefficient of 108.
• The constant terms, which are simple numbers without variables, also combine to form the final simplified expression.

When dealing with polynomials, always remember:

• Terms are grouped according to their variables and exponents.
• Operations follow the order of distribution and then combining like terms for simplification.

These expressions are foundational in algebra, leading to more complex operations and solutions.