Problem 24 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$4\left(-w^{3}-5 w^{2}+3 w+7\right) -\left(3 w^{3}-7 w^{2}+12 w+19\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-7w^3 - 13w^2 + 9\).

Step 1: Distribute the coefficients to each term in the brackets

First, distribute 4 to each term inside the first bracket and distribute -1 to each term inside the second bracket.

Step 2: Simplify each term inside the brackets

After distributing, we will multiply the coefficients with their corresponding terms: $$4(-w^3) - 4(5w^2) + 4(3w) + 4(7) - (3w^3 - 7w^2 + 12w + 19)$$

Step 3: Rewrite the expression

Now, rewrite the expression by removing the brackets and combining the terms: $$-4w^3 - 20w^2 + 12w + 28 - 3w^3 + 7w^2 - 12w -19$$

Step 4: Combine like terms

Combine the terms with the same degree of the variable w: \((-4w^3 - 3w^3) + (-20w^2 + 7w^2) + (12w - 12w) + (28 - 19)\)

Step 5: Simplify the combined terms

Simplify each group of like terms: \[-7w^3 - 13w^2 + 0w + 9\]

Step 6: Write the final simplified expression

Since the term \(\(0w\)\) does not contribute to the expression, we can remove it. The final simplified expression is: \[-7w^3 - 13w^2 + 9\]

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that helps to simplify expressions. It states that a coefficient multiplied by a term in parentheses must be distributed, or applied to each term within the parentheses.
For instance, if we have an expression like \(4(-w^3 - 5w^2 + 3w + 7)\), the "4" needs to be multiplied by each term inside the bracket:

• \(-4w^3\)
• \(-4 \, \times \, 5w^2\)
• \(4 \, \times \, 3w\)
• \(4 \, \times \, 7\)

The same applies to the second part \(-1(3w^3 - 7w^2 + 12w + 19)\), where "-1" must be multiplied by each term inside the parentheses. This property is essential for simplifying expressions and making them easier to understand and solve.

Combining Like Terms

After distributing, the next crucial step in simplifying polynomials is combining like terms. This involves looking for terms that have the same variable raised to the same power, known as the degree of the term.
For example, in the expression \(-4w^3 - 20w^2 + 12w + 28 - 3w^3 + 7w^2 - 12w - 19\), like terms are grouped and combined:

• \((-4w^3 - 3w^3)\) are combined as both are \(w^3\) terms.
• \((-20w^2 + 7w^2)\) are combined as both are \(w^2\) terms.
• \(12w - 12w\) combine to eliminate each other because their sum is zero.
• \((28 - 19)\) are constant terms that combine through simple subtraction.

Combining like terms simplifies the expression, making it less complex and easier to read.

Coefficients in Polynomials

A polynomial is an expression constructed from variables and coefficients. Coefficients are numeric values directly in front of variables that indicate how many times to multiply the variable. For example, in \(4w\), "4" is the coefficient.
In the expression resulting from our exercise, \(-7w^3 - 13w^2 + 9\), each term has a coefficient:

• \(-7\) is the coefficient of \(w^3\).
• \(-13\) is the coefficient of \(w^2\).
• \(9\) is the constant term and acts as a coefficient for \(w^0\), as it has no variable attached.

Understanding coefficients is key to performing operations on polynomials, such as addition, subtraction, and distribution, as they direct how we manage the variable parts during these operations.