Factorize k2 - 2kp + p2.

Analysis

The question asks us to factorize the quadratic expression $k^2 - 2kp + p^2$. This is a classic algebraic expression that follows the pattern of a Perfect Square Trinomial.

Concept

A perfect square trinomial follows the general form: $a^2 - 2ab + b^2 = (a - b)^2$

In this expression, we are looking for two terms that, when squared, give us the first and last terms of our expression, and when multiplied together and doubled, give us the middle term.

Step-by-Step Solution

1. Identify the terms: • The first term is $k^2$, which is $(k)^2$. • The last term is $p^2$, which is $(p)^2$. • The middle term is $-2kp$.

2. Check the middle term: • According to the formula $(a - b)^2 = a^2 - 2ab + b^2$, the middle term should be $-2 \times (\text{first term}) \times (\text{last term})$. • Here, $-2 \times k \times p = -2kp$. • Since this matches our middle term, the expression is indeed a perfect square.

3. Write the factors: • Since the middle term is negative, we use the subtraction form: $(k - p)^2$.

4. Verification: • $(k - p)(k - p) = k(k - p) - p(k - p)$ • $= k^2 - kp - kp + p^2$ • $= k^2 - 2kp + p^2$ • This confirms our factorization is correct.

Conclusion

Comparing our result $(k - p)^2$ with the given options, it matches option B.

Exam Tip

Whenever you see a quadratic expression with three terms where the first and last terms are perfect squares (like $x^2$ and $y^2$), always check if the middle term is exactly twice the product of the square roots of the first and last terms. If it is, you can immediately write it as $(a \pm b)^2$ without needing to go through the long process of splitting the middle term!