Problem 15 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 4 x^{2}+20 x $$

Step-by-Step Solution

Answer

Factor out 4x to get 4x(x + 5). x + 5 is a prime polynomial. Verification shows the factorization is correct.

Step 1: Identify the Greatest Common Factor (GCF)

Look at the terms in the polynomial: 4x^2 and 20x. The coefficients 4 and 20 have a GCF of 4. Both terms also have the variable x in common. Therefore, the GCF of the entire expression is 4x.

Step 2: Factor Out the GCF

Factor 4x out of each term in the expression: \[ 4x (x + 5) \]

Step 3: Identify Any Prime Polynomials

Check if the remaining polynomial inside the parentheses, x + 5, can be factored further. Since x + 5 is already in its simplest form and cannot be factored further, it is a prime polynomial.

Step 4: Check Your Work

To verify the factoring, distribute 4x back into the binomial: \[ 4x (x + 5) = 4x \times x + 4x \times 5 = 4x^2 + 20x \] The expression reverts to the original polynomial, confirming that the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

When factoring polynomials, the first step is to find the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in the polynomial. For the polynomial given, 4x² + 20x, we look at the coefficients and the variables:

• The coefficients are 4 and 20. The largest number that divides both is 4.
• Both terms include the variable x. So, x is also part of the GCF.

Combining these, the GCF for 4x² and 20x is 4x. Factoring out the GCF greatly simplifies the polynomial.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored further. To check if a polynomial is prime, try to break it down further using common factoring techniques. In this example:

• Once we factor out the GCF from 4x² + 20x, we're left with x + 5.
• Checking the polynomial x + 5, we see that there are no common factors or patterns (like the difference of squares, etc.)

This means x + 5 is already in its simplest form and is thus a prime polynomial.

Factoring Steps

Factoring polynomials involves specific steps to follow systematically:

• Step 1: Identify the GCF, which we've done as 4x for 4x² + 20x.
• Step 2: Factor the GCF out of each term. Thus, 4x² + 20x becomes 4x(x + 5).
• Step 3: Recognize any prime polynomials. Here, x + 5 is prime.

These steps ensure you're simplifying the polynomial correctly and efficiently.

Polynomial Verification

After factoring, it's crucial to verify the work. This confirmation ensures the polynomial is factored correctly. To verify:

• Distribute the GCF: Multiply 4x back into (x + 5).
• Perform the multiplication: 4x * x + 4x * 5 = 4x² + 20x.
• Check that the result matches the original polynomial.

In this case, 4x(x + 5) reverts to 4x² + 20x, confirming our factorization is correct.