Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x(x+3)-6(x+3) $$

Short answer

(x + 3)(x - 6)

Step-by-step solution

Identify Common Factors

Notice that both terms in the polynomial share a common factor of \(x + 3\).

Factor Out the Common Factor

Factor \(x + 3\) out of both terms: \(x(x + 3) - 6(x + 3) = (x + 3)(x - 6)\).

Verify the Factorization

Multiply the factors back together to ensure correctness: \((x + 3)(x - 6) = x(x - 6) + 3(x - 6) = x^2 - 6x + 3x - 18 = x^2 - 3x - 18\).\ Therefore, the factorization is correct.

Key concepts

common factors

Before diving into polynomial factorization, it is essential to understand the concept of common factors. A common factor is a number or expression that evenly divides two or more terms. Identifying common factors simplifies algebraic expressions and is the first crucial step in polynomial factorization.

In the exercise, we begin by looking for common factors in the polynomial \( x(x+3)-6(x+3) \). Both terms share a common factor of \( x+3 \). Recognizing this allows us to factor \( x+3 \) out, making the polynomial easier to manage.

Steps to identify common factors:

• Look at each term in the polynomial.
• Identify any factors that appear in every term.
• Factor these common elements out.

This process often simplifies even complex polynomials, laying the groundwork for further simplification.

polynomial factorization

Once you have identified common factors, the next step is polynomial factorization. Polynomial factorization involves rewriting a polynomial as a product of simpler polynomials that, when multiplied, give back the original polynomial. This step is crucial for solving polynomial equations and simplifying expressions.

In our exercise, after identifying \( x+3 \) as the common factor, we factor it out:

\( x(x + 3) - 6(x + 3) = (x + 3)(x - 6) \)

This result means we successfully factored the polynomial into two binomials, \((x + 3)\) and \((x - 6)\).

Steps to factorize a polynomial:

• Identify and factor out the greatest common factor (GCF).
• Rewrite the polynomial as a product of simpler terms.

Polynomial factorization simplifies solving and further manipulation of polynomials.

verifying factorization

Verifying factorization ensures that the factorization process has been done correctly. This step involves multiplying the factors back together to check if they reconstruct the original polynomial.

In the exercise, we verify the factorization as follows:

Multiply the factors: \((x + 3)(x - 6) = x(x - 6) + 3(x - 6)\)
Distribute and combine like terms: \(x^2 - 6x + 3x - 18 = x^2 - 3x - 18\)

Since these steps yield the original polynomial, our factorization is correct.

Steps to verify factorization:

• Multiply the factors back together.
• Simplify to see if you obtain the original polynomial.

Verification is an essential step, as it confirms the accuracy of the factorization process, ensuring no errors were made.