Problem 30 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$(s-5)\left(7 s^{2}-3 s-11\right)$$

Step-by-Step Solution

Answer

The simplified expression is: \(\boxed{7s^3 - 38s^2 + 4s + 55}\)

Step 1: Identify the Distributive Property

The Distributive Property states that: \[a(b + c) = ab + ac\] In this case, \(a = (s-5)\), \(b = 7s^2\), and \(c = -3s - 11\). We will perform the multiplication by following the distributive property for each term.

Step 2: Distribute the first polynomial

Now, distribute \((s-5)\) to each term in the trinomial expression \((7s^2 - 3s - 11)\): \((s-5)(7s^2 - 3s - 11) = (s-5)(7s^2) + (s-5)(-3s) + (s-5)(-11)\)

Step 3: Perform the multiplications

Perform the multiplication for each term: \[7s^2(s-5) = 7s^3 - 35s^2\] \[-3s(s-5) = -3s^2 + 15s\] \[-11(s-5) = -11s + 55\]

Step 4: Combine like terms

Now, combine all the terms obtained in the previous step: \((7s^3 - 35s^2) + (-3s^2 + 15s) + (-11s + 55)\) Combine the \(s^2\) terms and the \(s\) terms: \[7s^3 - 35s^2 - 3s^2 + 15s - 11s + 55\] \[7s^3 - 38s^2 + 4s + 55\] The simplified expression is: \[\boxed{7s^3 - 38s^2 + 4s + 55}\]

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of another polynomial. This road map allows us to break down complex expressions into simpler parts. When given two polynomials, like in the expression \((s-5)(7s^2 - 3s - 11)\), we apply the distributive property.

To multiply, you need to:

• Identify terms: Clearly outline all terms in each polynomial. Here, terms are \(s\) and \(-5\) for the first polynomial, and \(7s^2\), \(-3s\), and \(-11\) for the second polynomial.
• Apply distribution: Multiply each term in the first polynomial with each term in the second polynomial. Distribute \(s\) across all terms in the second polynomial, then distribute \(-5\) the same way.

This step-by-step multiplication ensures that no term is ignored, leading to the correct simplified form.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, making them easier to understand or solve. After multiplying polynomials, expressions can appear complex with multiple terms. To simplify, focus on rearranging and reducing expressions by following clear steps.

Steps to Simplify:

• Perform operations: Complete all multiplication and addition operations first to see all terms clearly. In our example, the individual multiplications yield \(7s^3 - 35s^2 - 3s^2 + 15s - 11s + 55\).
• Simplify numbers: If there are coefficients or constants that sum neatly, add or subtract them to reduce quantity of terms.

The goal is to have the expression in the simplest and most understandable format, reducing possibilities of mistakes in computation.

Combining Like Terms

Combining like terms is a critical step in simplifying expressions further. Like terms have the same variables raised to the same power, allowing them to be easily added or subtracted.

In our example, the expression \(7s^3 - 35s^2 - 3s^2 + 15s - 11s + 55\) seems cluttered at first glance.

• Identify like terms: Clearly identify and group terms having identical variable components, such as \(-35s^2\) and \(-3s^2\), or \(15s\) and \(-11s\).
• Combine coefficients: Only work with the coefficients of like terms. Add or subtract these coefficients while keeping the variable part unchanged. Here we find \(-35s^2 - 3s^2 = -38s^2\) and \(15s - 11s = 4s\).

This transforms the expression into a smoother, consolidated version: \(7s^3 - 38s^2 + 4s + 55\). Such simplification makes it easier to interpret results and proceed with further mathematical operations.