Problem 30 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ a^{5} b^{11}+a^{4} b^{10}+a^{3} b^{9}+a^{3} b^{5} $$

Step-by-Step Solution

Answer

Factor out \(a^{3}b^{5}\) to get \(a^{3}b^{5}(a^{2}b^{6} + ab^{5} + b^{4} + 1)\). The polynomial inside the parentheses is prime.

Step 1: Identify the Greatest Common Factor (GCF)

To factor out the GCF, first determine the lowest powers of each variable present in all terms. The terms are: 1. \(a^{5}b^{11}\)2. \(a^{4}b^{10}\)3. \(a^{3}b^{9}\)4. \(a^{3}b^{5}\). The lowest power of \(a\) is \(a^{3}\) and the lowest power of \(b\) is \(b^{5}\). Therefore, the GCF is \(a^{3}b^{5}\).

Step 2: Factor Out the GCF

Extract the GCF \(a^{3}b^{5}\) from each term:\[a^{5}b^{11} + a^{4}b^{10} + a^{3}b^{9} + a^{3}b^{5} = a^{3}b^{5}(a^{2}b^{6} + ab^{5} + b^{4} + 1)\].

Step 3: Identify Prime Polynomials

Examine the polynomial inside the parentheses, \(a^{2}b^{6} + ab^{5} + b^{4} + 1\). A polynomial is prime if it cannot be factored into polynomials of lower degree using integers. \(a^{2}b^{6} + ab^{5} + b^{4} + 1\) is already in its simplest form and is a prime polynomial.

Step 4: Check the Work

Multiply the factored form back to its original expression to ensure correctness:\[a^{3}b^{5}(a^{2}b^{6} + ab^{5} + b^{4} + 1) = a^{5}b^{11} + a^{4}b^{10} + a^{3}b^{9} + a^{3}b^{5}\].

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as the product of its greatest common factor and another polynomial with smaller degrees.
In the given exercise, we first identified the greatest common factor (GCF) from the polynomial terms:

• \(a^{5}b^{11}\)
• \(a^{4}b^{10}\)
• \(a^{3}b^{9}\)
• \(a^{3}b^{5}\)

Factoring out the GCF (the lowest powers of common variables), \(a^{3}b^{5}\), we factored the original polynomial to get:
\a^{5}b^{11} + a^{4}b^{10} + a^{3}b^{9} + a^{3}b^{5} = a^{3}b^{5}(a^{2}b^{6} + ab^{5} + b^{4} + 1)\
This process simplifies the polynomial and can make further operations easier.

Prime Polynomials

A prime polynomial is a polynomial that cannot be factored further using integers.
After factoring out the GCF from the given polynomial, we were left with \(a^{2}b^{6} + ab^{5} + b^{4} + 1\).
To check if this polynomial is prime, we examined if it could be factored into two polynomials of lower degrees.
Since it cannot be factored further, we conclude that \(a^{2}b^{6} + ab^{5} + b^{4} + 1\) is a prime polynomial.
Recognizing prime polynomials is crucial because it simplifies the problem and avoids unnecessary calculations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations.
In the exercise, we dealt with an algebraic expression involving powers of variables \(a\) and \(b\):
\a^{5}b^{11} + a^{4}b^{10} + a^{3}b^{9} + a^{3}b^{5}\
Working with algebraic expressions involves

• Identifying common factors
• Factoring to simplify expressions
• Checking for prime conditions

Understanding the structure of algebraic expressions helps in simplifying and solving equations effectively.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial.
In the checking step, we multiplied the factored form back to its original form.
The expression used in the exercise was:
\a^{3}b^{5}(a^{2}b^{6} + ab^{5} + b^{4} + 1)\
Using distributive property, each term inside the parentheses is multiplied by \(a^{3}b^{5}\):

• \(a^{3}b^{5} \times a^{2}b^{6} = a^{5}b^{11}\)
• \(a^{3}b^{5} \times ab^{5} = a^{4}b^{10}\)
• \(a^{3}b^{5} \times b^{4} = a^{3}b^{9}\)
• \(a^{3}b^{5} \times 1 = a^{3}b^{5}\)

This verifies the factorization and ensures accuracy.
Each step consolidates the concept of multiplying polynomials and reinforces the importance of the distributive property in algebra.