Problem 87 - Introductory Algebra for College Students

Question

Factor completely. $$-32 x^{2} y^{4}+20 x y^{4}+12 y^{4}$$

Step-by-Step Solution

Answer

The completely factored form of the expression \(-32x^{2}y^{4}+20xy^{4}+12y^{4}\) is \(4y^{4}(-8x^{2}+5x+3)\).

Step 1 - Identify the Common Factor

Observe the equation \( -32x^{2}y^{4}+20xy^{4}+12y^{4}\) and identify the common factors. In this case, \(4y^{4}\) is a term that is common to all the coefficients.

Step 2 - Factor Out the Common Factor

Factor out the common factor \(4y^{4}\) from the equation. The equation becomes: \(4y^{4}(-8x^{2}+5x+3)\).

Step 3 - Check for Further Factoring

Verify if the polynomial \(-8x^{2}+5x+3\) within the brackets can be factored further. As we observe, this cannot be factored further using integers as it does not factor cleanly.

Key Concepts

Understanding the Common Factor

In polynomial factoring, the first and crucial step is to identify the common factor. This factor is a number or a term that appears in each term of the polynomial. By pulling this out, you simplify the polynomial. In the exercise provided, all terms in the polynomial -32x²y⁴ + 20xy⁴ + 12y⁴share a common factor of \(4y^4\). When looking for a common factor, focus on:

• Coefficients: Find the greatest common divisor (GCD). For -32, 20, and 12, it is 4.
• Variables: Look for the lowest power of common variables. Here, \(y^4\) is common across all terms.

By extracting \(4y^4\), you simplify the polynomial and make subsequent steps easier. This step lays the foundation for further factoring or recognizing that the polynomial is fully factored.

Diving into Factoring Techniques

Factoring techniques in polynomials involve various strategies to simplify or resolve expressions. Once you have identified and extracted a common factor, as in the exercise with the polynomial -32x²y⁴ + 20xy⁴ + 12y⁴, you are left with a simpler form now noted as:\[4y^4(-8x^2 + 5x + 3)\]Here are some primary factoring techniques:

• **Factoring by grouping**: This involves grouping terms to factor common elements within each group.
• **Using special formulas**: Sometimes recognizing patterns like perfect squares or cubes helps in factoring.
• **Trial and error**: For quadratic polynomials, like the expression within the parentheses above, try applying the FOIL method to see if it factors further.

Not all polynomials factor neatly, as seen in our expression -8x² + 5x + 3, which does not break down further using integer factoring techniques.

Exploring Polynomials

Polynomials are algebraic expressions made up of variables and coefficients, where the operations involve addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, in the expression\(-32x^{2}y^{4}+20xy^{4}+12y^{4}\),we observe:

• **Terms**: Separate components like \(-32x^{2}y^{4}\), each often containing variables and coefficients.
• **Degree**: Determined by the highest power of the variable. Here, the degree in terms of \(x\) is 2, and for \(y\) it is 4.
• **Monomials, binomials, and polynomials**: These describe one-term, two-term, and multi-term expressions respectively.

Grasping what defines a polynomial helps in systematically approaching factoring by understanding its structure and the potential for simplification.