(x-a)(x-b)(x-c)(x-d)... (x-z)

4 min read Jul 03, 2024

Polynomial Expansion: Understanding the Product of Linear Factors

In algebra, when we are faced with a product of linear factors in the form of (x-a)(x-b)(x-c)(x-d)...(x-z), it can be a daunting task to expand it into a polynomial expression. However, with a solid understanding of the concept and a few simple rules, we can master this skill.

What is the Product of Linear Factors?

The product of linear factors is an expression consisting of two or more linear factors multiplied together. Each linear factor is a binomial of the form x - k, where k is a constant. For example, (x-2)(x-3) is a product of two linear factors.

Expanding the Product

To expand the product of linear factors, we need to follow the distributive property of multiplication over addition. This means we need to multiply each term in the first factor by each term in the second factor, and so on.

Let's consider a simple example: (x-2)(x-3). To expand this, we need to multiply each term in the first factor by each term in the second factor:

x in the first factor multiplied by x in the second factor gives x^2
x in the first factor multiplied by -3 in the second factor gives -3x
-2 in the first factor multiplied by x in the second factor gives -2x
-2 in the first factor multiplied by -3 in the second factor gives 6

Combining these terms, we get the expanded polynomial: x^2 - 5x + 6.

Expanding a Longer Product

Now, let's consider a longer product of linear factors: (x-a)(x-b)(x-c)(x-d)...(x-z). To expand this, we need to follow the same rules as before, but with more factors.

Start by multiplying the first two factors: (x-a)(x-b) = x^2 - (a+b)x + ab
Multiply the result by the third factor: (x^2 - (a+b)x + ab)(x-c) = x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc
Continue this process until you have multiplied all the factors together.

As you can see, the process can get quite lengthy and prone to errors. However, with practice and patience, you can master the art of expanding products of linear factors.

Conclusion

In conclusion, expanding the product of linear factors is an essential skill in algebra. By following the distributive property and multiplying each term in each factor, we can expand the product into a polynomial expression. Remember to be patient and take your time when working with longer products, and you'll become proficient in no time!