Factorise fully the following:
a) x² + 3x
The Correct answer and Explanation is:
To factor the expression x2+3xx^2 + 3xx2+3x fully, we first identify any common factors in the terms of the polynomial.
Step 1: Identify the Common Factor
The given expression is:x2+3xx^2 + 3xx2+3x
We can see that both terms, x2x^2×2 and 3x3x3x, have a common factor of xxx.
Step 2: Factor Out the Common Factor
Next, we factor out the common factor xxx:x2+3x=x(x+3)x^2 + 3x = x(x + 3)x2+3x=x(x+3)
Step 3: Verify the Factorization
To ensure that our factorization is correct, we can expand the expression x(x+3)x(x + 3)x(x+3):x(x+3)=x2+3xx(x + 3) = x^2 + 3xx(x+3)=x2+3x
This confirms that we have factored the expression correctly since the expanded form matches the original expression.
Explanation of the Process
- Identifying Common Factors: In polynomial expressions, looking for common factors is often the first step in factorization. The term x2x^2×2 indicates that xxx is a factor of the entire expression. Similarly, 3x3x3x also contains xxx as a factor.
- Factoring Out: Once we identify the common factor, we can factor it out from each term. In this case, factoring xxx out of both terms leaves us with the binomial (x+3)(x + 3)(x+3).
- Re-checking: After performing the factorization, it’s essential to expand the factors back to verify that we return to the original expression. This step helps ensure that no mistakes were made during the factorization process.
Final Answer
The fully factored form of the expression x2+3xx^2 + 3xx2+3x is:x(x+3)\boxed{x(x + 3)}x(x+3)
By following these steps, we can factor many polynomials and confirm their accuracy. This method highlights the importance of recognizing patterns and simplifying expressions through common factors, which is a fundamental skill in algebra.