Problem 104 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$-10 a^{4} b^{2}+15 a^{3} b^{3}+25 a^{2} b^{4}$$

Step-by-Step Solution

Answer

The factorized form of the given polynomial is \(5a^{2}b^{2}(-2a^{2}+3ab+5b^{2})\).

Step 1: Identify the common factors

The first step is to identify common factors in each term of the polynomial. In this case, the common factors are \(a^2\), \(b^2\), and \(5\) from all three terms of the given polynomial.

Step 2: Factorize the polynomial

Remove these common factors from each term in the polynomial, resulting in the expression \(5a^{2}b^{2}(-2a^{2}+3ab+5b^{2})\)

Step 3: Verify through multiplication

To ensure that the factorization is correct, multiply the factored terms. If the result matches the original, the factorization is correct. Therefore, we multiply the factored expression \(5a^{2}b^{2}(-2a^{2}+3ab+5b^{2})\) and see that it results in the original polynomial \(-10a^{4}b^{2}+15a^{3}b^{3}+25a^{2}b^{4}\).

Key Concepts

Polynomials in Several Variables

Polynomials in several variables are like regular polynomials, but they include more than one variable. In the expression \(-10 a^{4} b^{2}+15 a^{3} b^{3}+25 a^{2} b^{4}\), there are two variables: \(a\) and \(b\). These polynomials represent expressions where each term is a product of coefficients and the powers of these variables.

Understanding how to manage these can be crucial, especially in factoring, as each variable might have different powers. Here’s what you need to know:

• Every term in the polynomial can contain different powers of each variable.
• When factoring, it is important to look at each variable and its powers separately to identify common factors.

Polynomials in several variables add complexity due to the interleaving terms, but by breaking down each component, they can be managed efficiently.

Common Factors

A common factor is a value or expression that divides each term of the polynomial without a remainder. When it comes to polynomials in several variables, identifying common factors is the key to successful factorization.

In our example, common factors that appear in every term are \(a^2\), \(b^2\), and \(5\).

Here's how to identify them:

• Look for the highest power of every variable that appears in all terms. In the example, \(a^2\) and \(b^2\) were common across all terms.
• Identify any numerical coefficients that can divide all the terms evenly. Here, \(5\) is a common factor.

By extracting these common factors, you simplify the polynomial, preparing it for further manipulation or verification.

Multiplication Verification

After you factor a polynomial, it’s crucial to verify the accuracy of your factorization. This is done through multiplication verification.

To verify, multiply back the factors you have extracted. For the example \(5a^{2}b^{2}(-2a^{2}+3ab+5b^{2})\), perform the multiplication:

• Distribute each term in the binomial \((-2a^{2} + 3ab + 5b^{2})\) by the common factors \(5a^2b^2\).
• Check if the result equals the original polynomial \(-10a^{4}b^{2}+15a^{3}b^{3}+25a^{2}b^{4}\).

If it does match, you have successfully verified your factorization. This step ensures that no mistakes were made, giving you confidence in your result.