Problem 68 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 7 a-7 b^{2} $$

Step-by-Step Solution

Answer

7(a - b^2); the polynomial inside the parentheses is prime.

Step 1: Identify the common factor

Observe that both terms in the expression have a common factor. In this case, 7 is the common factor.

Step 2: Factor out the common factor

Extract the common factor from each term in the expression. Factoring out 7 from the terms, we get:\[ 7a - 7b^2 = 7(a - b^2) \]

Step 3: Check if remaining polynomial is prime

Examine the remaining polynomial inside the parentheses. The expression \( a - b^2 \) cannot be factored further using integers, so it is considered a prime polynomial.

Key Concepts

Common Factors

In algebra, finding common factors is a crucial first step in polynomial factorization. Common factors are numbers or variables that are shared by all terms in an expression. In our example, the polynomial is 7a - 7b^2. Both terms, 7a and 7b^2, share the common factor of 7. Common factors can make complex polynomials easier to manage and simplify.

To identify common factors, follow these steps:

• Look at each term in the polynomial.
• Identify any constants (numbers) and variables that appear in every term.
• Factor them out by dividing each term by the common factor.

As a general rule, this process helps simplify polynomials and reveals potential prime polynomials or further factorable expressions.

Prime Polynomial

A prime polynomial is similar to a prime number in that it cannot be factored further using integers. Once you have factored out the common factors, you need to check if the remaining polynomial can be factorized further.

In our example, after factoring out the 7, we are left with a - b^2. This new expression inside the parentheses is what we'll examine next.

• If the polynomial inside the parenthesis can no longer be broken down or simplified using integer factors, it is considered a prime polynomial.
• In this case, a - b^2 does not have any common factors or further factorization options using integers.

Therefore, a - b^2 is a prime polynomial. Recognizing prime polynomials is essential as it determines whether you need to continue factoring or if you've reached the simplest form.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler parts called factors. This technique can be useful in solving equations more easily.

Here are general steps to follow when factoring polynomials:

• Identify and factor out any common factors from the entire polynomial.
• Examine the reduced polynomial to see if it can be further factored.
• If the remaining polynomial is prime, then the factorization process is complete.

In our exercise, we started with the polynomial 7a - 7b^2. By factoring out the common factor of 7, we simplified it to 7(a - b^2). Since a - b^2 is a prime polynomial, our factorization is complete.

Understanding and mastering polynomial factorization can make complex algebraic problems easier to solve and help identify relationships between different polynomials in mathematical problems.