Problem 82 - Elementary and Intermediate Algebra

Question

Factor by grouping. $$ 9 s^{3}-2 s^{2}+36 s-8 $$

Step-by-Step Solution

Answer

The polynomial factors as \((s^2 + 4)(9s - 2)\).

Step 1: Split the Polynomial

To factor by grouping, first split the polynomial into two pairs: \( (9s^3 - 2s^2) + (36s - 8) \).

Step 2: Factor out the Greatest Common Factor (GCF) from Each Pair

Find the GCF of each pair. For the first pair \(9s^3 - 2s^2\), the GCF is \(s^2\). For the second pair \(36s - 8\), the GCF is \(4\). This gives us:\[ s^2(9s - 2) + 4(9s - 2) \]

Step 3: Apply Factor by Grouping

Notice that \((9s - 2)\) is a common factor in both terms. We can factor \((9s - 2)\) out of the entire expression:\[ (s^2 + 4)(9s - 2) \]

Step 4: Verify the Factorization

Distribute to check the factorization: Expand \((s^2 + 4)(9s - 2)\) to see if it returns to the original polynomial:\( s^2(9s) + s^2(-2) + 4(9s) + 4(-2) \) simplifies to \(9s^3 - 2s^2 + 36s - 8\).

Step 5: Confirm the Final Factorization

The final factorization is correctly confirmed as:\[ (s^2 + 4)(9s - 2) \] due to the matching with the original polynomial: \( 9s^3 - 2s^2 + 36s - 8 \).

Key Concepts

Polynomial Grouping

Factoring polynomials by grouping is a clever way to break down complex expressions into simpler parts. This method is particularly useful when dealing with polynomials that cannot be easily factored using standard methods. The technique involves splitting the polynomial into smaller groups, making it easier to identify common factors. In the given exercise, the polynomial \(9s^3 - 2s^2 + 36s - 8\) is split into two pairs:

• \((9s^3 - 2s^2)\)
• \((36s - 8)\)

This division into groups allows us to consider each pair separately and generate a more manageable path for simplification. Start by organizing terms that have common factors together. Grouping doesn't change the overall value of the expression but makes it easier to visualize and manipulate.

Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is crucial in the process of factoring by grouping. The GCF is the largest factor that divides each term of the polynomial completely. For each grouped pair in our polynomial, identify and extract this common factor.For the first group \(9s^3 - 2s^2\), the GCF is \(s^2\). For the second pair \(36s - 8\), the GCF is \(4\). Once we factor out these GCFs, the expression becomes:\[ s^2(9s - 2) + 4(9s - 2) \]This step of factoring out the GCF simplifies the polynomial, revealing common expressions that can be further factored. The GCF helps to reduce the complexity of each pair, thus revealing patterns or repeated factors within the polynomial that we can use for further simplification.

Expression Simplification

Expression simplification involves using factors to condense an expression into a more straightforward form. In this exercise, once both pairs have been factored, notice that \(9s - 2\) appears as a common factor in both terms:

• \(s^2(9s - 2)\)
• \(4(9s - 2)\)

By extracting this common factor, the entire expression can be rewritten more simply as:\[ (s^2 + 4)(9s - 2) \]This highlights the elegance of expression simplification, reducing a seemingly complex polynomial into more basic components. Finally, verify the accuracy of the simplification by distributing the factors to ensure that it results in the original polynomial. Through this process, expression simplification provides clarity and transforms the polynomial into its most comprehensible form.