Problem 23 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 15 a^{3} b^{6}-90 a b^{7}+35 a^{2} b^{5} $$

Step-by-Step Solution

Answer

GCF is \(5a b^5\); factored form: \[5a b^5 (3a^2b - 18b^2 + 7a)\]. Remaining polynomial is prime.

Step 1 - Identify the greatest common factor (GCF)

First, determine the greatest common factor of the coefficients (15, 90, and 35), which is 5. Then, identify the common variable terms with the lowest powers: for the variable part, we have at least one 'a' and five 'b's across all terms. Thus, the GCF is \(5a b^5\).

Step 2 - Factor out the GCF

Factor the GCF from each term in the polynomial: \[15a^3b^6 - 90ab^7 + 35a^2b^5 = 5a b^5 (3a^2b - 18b^2 + 7a)\].

Step 3 - Identify prime polynomials

Examine the remaining polynomial \(3a^2b - 18b^2 + 7a\). It cannot be factored further using integer coefficients, so it is considered a prime polynomial.

Step 4 - Check the factorization

Multiply the factored terms to ensure accuracy: \[5a b^5 (3a^2b - 18b^2 + 7a) = 15a^3 b^6 - 90a b^7 + 35a^2 b^5\]. Since we obtain the original polynomial, the factorization is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is essential in algebra as it simplifies expressions, making them easier to work with. A polynomial is an expression consisting of variables, coefficients, and exponents. To factor a polynomial, you start by identifying the greatest common factor (GCF) among the terms. The GCF is the highest number and common variable factor that divides each term in the polynomial.

Taking the example:

\(15a^3b^6 - 90ab^7 + 35a^2b^5\)

The steps for factoring include:

• Identify the GCF of the coefficients (15, 90, and 35), which is 5.
• Determine the lowest power of common variables (a^1 and b^5).

Putting these together, the GCF of this polynomial is \(5a b^5\).

Now, factor this GCF out:

\(15a^3b^6 - 90ab^7 + 35a^2b^5 = 5a b^5 (3a^2b - 18b^2 + 7a)\)

The polynomial is successfully factored, simplifying further operations.

Prime Polynomial

A prime polynomial is one that cannot be factored any further using integer coefficients. After factoring out the GCF from our example, we are left with:

\(3a^2b - 18b^2 + 7a\)

To determine if this is a prime polynomial, we attempt to factor it. Since there are no integer factors that multiply to obtain this polynomial, it remains unfactorable by integers and hence is classified as a prime polynomial.

Prime polynomials hold a similar concept to prime numbers—they are indivisible by anything other than 1 and themselves within the same domain of integers. Recognizing prime polynomials is useful for knowing when a polynomial has been simplified as much as possible.

Variable Exponent

Understanding variable exponents is key when working on polynomial factorization. Exponents indicate how many times a variable is multiplied by itself.

• The exponent in <\(a^3\)> means 'a' is multiplied by itself three times: a * a * a.
• The exponent in <\(b^6\)> means 'b' is multiplied by itself six times: b * b * b * b * b * b.

In the polynomial

\(15a^3b^6 - 90ab^7 + 35a^2b^5\),

each term contains variables raised to certain exponents. Finding the GCF involves identifying the variable terms with the lowest exponents common to each term. Here, the lowest power of 'a' common in all terms is a^1, and for 'b' it is b^5.

So, the GCF includes

\(a^1 b^5\).

Variable exponents help describe the degree of each term in a polynomial, guiding algebraic operations like factoring.