Problem 50 - Algebra A Combined Function

Question

Multiply. \((3+b)\left(2-5 b-3 b^{2}\right)\)

Step-by-Step Solution

Answer

The expanded expression is \(-3b^3 - 14b^2 - 13b + 6\).

Step 1: Distribute the First Term

We begin the multiplication by distributing the first term of the first polynomial, which is 3, across each term of the second polynomial. This gives us:\[ 3(2 - 5b - 3b^2) = 6 - 15b - 9b^2 \]

Step 2: Distribute the Second Term

Next, distribute the second term of the first polynomial, which is \(b\), across each term of the second polynomial:\[ b(2 - 5b - 3b^2) = 2b - 5b^2 - 3b^3 \]

Step 3: Combine Like Terms

Now, combine all terms we obtained from Steps 1 and 2 into a single expression and then group and add like terms:\[ 6 - 15b - 9b^2 + 2b - 5b^2 - 3b^3 = 6 - 13b - 14b^2 - 3b^3 \]

Step 4: Write the Final Expression

The final step is to write out the simplified expression as the final answer:\[ -3b^3 - 14b^2 - 13b + 6 \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental algebraic concept used to simplify expressions involving polynomial multiplication. It allows us to distribute a single term across a sum or difference. In this exercise, we apply this property twice.

Let's break it down: when we have an expression like \(a(b+c)\), we multiply \(a\) by each term inside the parentheses, leading us to \(ab + ac\).

- **Example:** For \(3(2 - 5b - 3b^2)\), we apply this property by multiplying \(3\) with each term inside: \(6 - 15b - 9b^2\). - For \(b(2 - 5b - 3b^2)\), we distribute \(b\), resulting in \(2b - 5b^2 - 3b^3\).

Mastering the distributive property is essential for tackling more complex polynomial equations with confidence.

Combining Like Terms

After applying the distributive property, the next crucial step is combining like terms. This step involves adding or subtracting terms that have the exact same variable raised to the same power.

Let's see how it works in our specific example. Once we have expanded the expression using the distributive property, we get a series of terms:

• From the distribution of \(3\): \(6 - 15b - 9b^2\)
• From the distribution of \(b\): \(2b - 5b^2 - 3b^3\)

Now, we need to make the expression simpler by grouping terms with similar variables and adding them together:

• Combine \(-15b\) and \(+2b\) to get \(-13b\).
• Combine \(-9b^2\) and \(-5b^2\) to get \(-14b^2\).

This results in a more concise expression: \(-3b^3 - 14b^2 - 13b + 6\).
Effectively combining like terms simplifies calculations and paves the way for easier problem-solving.

Polynomial Expression Simplification

Simplifying polynomial expressions is all about making them as straightforward as possible by combining like terms and arranging terms in descending powers. This strategy is demonstrated wonderfully in this exercise.

The expression \(6 - 15b - 9b^2 + 2b - 5b^2 - 3b^3\) starts out a bit messy. Hence, simplifying it becomes crucial.

- First, we grouped like terms:
- Terms involving \(b^3\), \(-3b^3\) - Terms involving \(b^2\), which combine to \(-14b^2\) - Terms involving \(b\), which combine to \(-13b\) - The constant term remains \(+6\)

• This gives us the neat and simplified polynomial: \(-3b^3 - 14b^2 - 13b + 6\).

By understanding the simplification process, you can bring polynomials to a form that's easier to interpret and solve.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division. In polynomial multiplication, we frequently use these operations to manipulate expressions. In this task, we focused on multiplication and the addition/subtraction involved in combining like terms.

Understanding how to apply these operations correctly is key to solving equations efficiently. Let's review the primary operations used:

• **Multiplication**: Used in distributing terms across other terms, showcasing skills in expanding brackets (i.e., \(3(2) = 6\)).
• **Addition/Subtraction**: Essential for combining like terms, where you add or subtract coefficients leveraging the same power of variables (i.e., \(-15b + 2b = -13b\)).

These core algebraic operations can transform complex expressions into simpler forms, allowing for a deeper understanding and further manipulation in advanced problem-solving situations.