Problem 106 - Elementary Algebra

Question

Factor out the GCF first and then factor by grouping. $$ -9 x 4+6 x 3-45 x 2+30 x $$

Step-by-Step Solution

Answer

The expression factors to \(3x(x^2 + 5)(-3x + 2).\)

Step 1: Identify the Greatest Common Factor (GCF)

Examine each term in the expression \(-9x^4 + 6x^3 - 45x^2 + 30x\) to find the greatest common factor. Each term is divisible by 3x, so the GCF is 3x.

Step 2: Factor Out the GCF

Factor out the GCF, 3x, from each term in the expression: \(3x(-3x^3 + 2x^2 - 15x + 10)\).

Step 3: Group the Terms

Divide the expression inside the parentheses \(-3x^3 + 2x^2 - 15x + 10\) into two groups: \((-3x^3 + 2x^2)\) and \((-15x + 10)\).

Step 4: Factor out the GCF from Each Group

Factor out the GCF for each group. For \((-3x^3 + 2x^2),\) factor out \(x^2\) to get \(x^2(-3x + 2)\). For \((-15x + 10),\) factor out \(5\) to get \(5(-3x + 2)\).

Step 5: Factor by Grouping

Because each group contains \(-3x + 2,\) factor \(-3x + 2\) from each expression to get: \((x^2 + 5)(-3x + 2).\)

Step 6: Write the Final Factored Expression

Combine all parts to write the completely factored form: \(3x(x^2 + 5)(-3x + 2).\)

Key Concepts

Greatest Common Factor (GCF)

To factor algebraic expressions efficiently, we begin by identifying the Greatest Common Factor (GCF). The GCF is the largest algebraic term that can divide each term of the expression without leaving a remainder. This step simplifies the expression, making further factoring steps much easier.

For instance, in the expression \(-9x^4 + 6x^3 - 45x^2 + 30x\), each term can be divided by \("3x"\). Thus, the GCF here is \(3x\). Finding the GCF involves determining both the highest number and the highest power of variables present that divide each term.

• Look for the smallest exponent in each variable.
• Identify the largest number that divides all coefficients.

Once the GCF is factored out, the expression greatly simplifies and prepares it for further methods like grouping or advanced polynomial techniques.

Factor by Grouping

Factor by grouping is a technique used when dealing with polynomials that have terms with common factors, but no overarching GCF for all terms. The steps for factor by grouping involve splitting the polynomial into smaller groups, making them easier to manage.

• Take the reduced polynomial inside the parentheses after factoring out the GCF.
• Divide the polynomial into two pairs or groups of terms.

In our exercise's expression \(-3x^3 + 2x^2 - 15x + 10\), you would initially group them into \((-3x^3 + 2x^2)\) and \((-15x + 10)\).

The next step is to find the GCF of each group. For example, the GCF of \(-3x^3 + 2x^2\) is \(x^2\), resulting in \(x^2(-3x + 2)\). Similarly, for \(-15x + 10\), it is \(5\), forming \(5(-3x + 2)\). With both groups having a common factor \(-3x + 2\), you can now combine them neatly by factoring this term out.

Polynomial Factoring

Once you complete the steps of factoring the GCF and utilizing factor by grouping, you arrive at polynomial factoring. This fundamental algebraic technique simplifies polynomials into products of smaller degree polynomials, thereby solving higher degree equations more efficiently.

From our exercise involving \(3x(-3x^3 + 2x^2 - 15x + 10)\), through factoring by grouping, results in \(3x(x^2 + 5)(-3x + 2)\). This is the final factored form. You have transformed the original polynomial into a product of linear and quadratic factors.

• Factoring completely makes it easier to find roots or solutions of the polynomial equation.
• Understanding each step ensures proper and complete factorization.

Remember, polynomial factoring is not merely about division; it's about strategic dismantling of the equation into more manageable expressions.