Problem 32 - Algebra and Trigonometry

Question

29-34 . Factor the expression by grouping terms. $$ -9 x^{3}-3 x^{2}+3 x+1 $$

Step-by-Step Solution

Answer

The expression factors to \((3x + 1)(-3x^2 + 1)\).

Step 1: Group Terms

Take the given expression \(-9x^3 - 3x^2 + 3x + 1\) and separate it into two groups. Group the first two terms together and the last two terms together: \((-9x^3 - 3x^2) + (3x + 1)\).

Step 2: Factor Out the Greatest Common Factor from Each Group

From the first group \(-9x^3 - 3x^2\), factor out \(-3x^2\) as it is the greatest common factor: \(-3x^2(3x + 1)\). From the second group \(3x + 1\), there is no common factor other than 1, so it stays as \(3x + 1\). The expression now looks like this: \(-3x^2(3x + 1) + 1(3x + 1)\).

Step 3: Factor by Grouping

Notice that \((3x + 1)\) is common in both terms \(-3x^2(3x + 1)\) and \(+ 1(3x + 1)\). Factor \((3x + 1)\) out: \((3x + 1)(-3x^2 + 1)\).

Step 4: Final Review

Check that each term was grouped and factored correctly. The expression \(-9x^3 - 3x^2 + 3x + 1\) has been factored successfully into \((3x + 1)(-3x^2 + 1)\). No further simplification can be achieved here.

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is fundamental in polynomial factoring. It refers to the largest factor that divides each term of a given expression completely. Identifying the GCF is the first step in simplifying or factoring expressions.
This process involves:

• Listing the factors of each term.
• Determining the common factors among these lists.
• Choosing the greatest one among the common factors.

For example, in the expression \[-9x^3 - 3x^2\], the GCF is \(-3x^2\).
By factoring it out, you simplify the expression to \(-3x^2(3x + 1)\).
The GCF allows for breaking down expressions into more manageable parts.

Expression Simplification

Expression simplification involves reducing a polynomial to its simplest form while maintaining its value.
In polynomial expressions, like in the given example \(-9x^3 - 3x^2 + 3x + 1\), simplification often uses techniques such as factoring by grouping.
This means:

• Splitting terms into smaller groups that can individually be factored.
• Factoring out the GCF from each group, if possible.
• Looking for commonalities between these grouped expressions.

By simplifying, we move from a complex expression to a simpler factored form, which in this case becomes \((3x + 1)(-3x^2 + 1)\).
This reformulation often reveals simpler relationships between variables or exposes the roots of polynomial equations.

Polynomial Factoring

Polynomial factoring is a method used to express the polynomial as a product of its factors.
The process of factoring by grouping is especially useful when dealing with four-term polynomials.
Here’s how it typically works:

• Identify groups within the polynomial that share a common factor.
• Factor out these common elements from each group.
• Observe for a common binomial factor within the newly created expressions.
• Factor out the common binomial to rewrite the polynomial as a product of factors.

For the expression \(-9x^3 - 3x^2 + 3x + 1\), we grouped and factored to form:\((3x + 1)(-3x^2 + 1)\).
This approach not only helps in simplifying the expressions but also in solving polynomial equations or finding zeros of the function efficiently.