(x-b)(x-c)(x-d)-formula.jpeg "(x-a)(x-b)(x-c)(x-d) Formula")
The General Formula for Expanding (x-a)(x-b)(x-c)(x-d)
In algebra, expanding the product of multiple binomials is a crucial skill. One such product is (x-a)(x-b)(x-c)(x-d), which can be expanded into a polynomial of degree 4. In this article, we will explore the general formula for expanding this product.
The Formula
The formula for expanding (x-a)(x-b)(x-c)(x-d) is:
(x-a)(x-b)(x-c)(x-d) = x^4 - (a+b+c+d)x^3 + (ab+ac+ad+bc+bd+cd)x^2 - (abc+abd+acd+bcd)x + abcd
This formula can be proven by multiplying out the binomials step by step, but it's often more convenient to memorize it or use it as a reference.
How to Apply the Formula
To apply the formula, simply replace a, b, c, and d with the desired values. For example, if we want to expand (x-2)(x-3)(x-4)(x-5), we would plug in a=2, b=3, c=4, and d=5 to get:
(x-2)(x-3)(x-4)(x-5) = x^4 - (2+3+4+5)x^3 + (2*3+2*4+2*5+3*4+3*5+4*5)x^2 - (2*3*4+2*3*5+2*4*5+3*4*5)x + 2*3*4*5
Simplifying this expression, we get:
(x-2)(x-3)(x-4)(x-5) = x^4 - 14x^3 + 76x^2 - 150x + 120
Conclusion
The formula for expanding (x-a)(x-b)(x-c)(x-d) is a powerful tool for simplifying complex algebraic expressions. By memorizing this formula, you can quickly expand products of binomials and solve a wide range of algebraic problems.
