Problem 18 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 21 p-56 w $$

Step-by-Step Solution

Answer

GCF is 7, factored form is \(7(3p - 8w)\), and \(3p - 8w\) is prime.

Step 1 - Identify the Greatest Common Factor (GCF)

Determine the GCF of the coefficients 21 and 56. The GCF is 7 because 7 is the largest number that can divide both 21 and 56.

Step 2 - Factor Out the GCF

Factor out the GCF (7) from the expression. This means we divide each term by 7 and place the 7 outside a set of parentheses: \(21p - 56w = 7(3p - 8w)\).

Step 3 - Identify Prime Polynomials

Check whether the polynomial inside the parentheses, \(3p - 8w\), can be factored further. Since 3 and 8 have no common factors other than 1, and the polynomial is already in its simplest form, \(3p - 8w\) is a prime polynomial.

Step 4 - Verify the Factoring

Multiply the factored expression back to confirm that it returns the original expression: \[ 7(3p - 8w) = 21p - 56w \]. The original expression is recovered, confirming that the factoring is correct.

Key Concepts

greatest common factor

To start factoring a polynomial, the first step is to identify the Greatest Common Factor (GCF). The GCF is the largest number that can divide each term in the polynomial without leaving a remainder. This helps simplify the polynomial by