(x-b)(x-c)-formula.jpeg "(x-a)(x-b)(x-c) Formula")
Understanding the (x-a)(x-b)(x-c) Formula
The formula (x-a)(x-b)(x-c) is a fundamental concept in algebra, particularly when dealing with polynomial factorization and solving cubic equations. This formula represents a cubic polynomial, which is a polynomial of degree 3.
What does it represent?
This formula represents a cubic polynomial with roots at x = a, x = b, and x = c. In simpler terms, it means that if you substitute any of these values for x, the entire expression will equal zero.
Expanding the Formula
You can expand the formula to get the standard form of the cubic polynomial:
(x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc
This expanded form is useful for:
- Finding the coefficients of the polynomial: The coefficients of the polynomial are directly related to the sum and product of the roots.
- Recognizing cubic polynomials: If you see a cubic polynomial in its standard form, you can potentially factor it using the (x-a)(x-b)(x-c) formula if you can identify the roots.
Using the Formula to Solve Cubic Equations
The formula can be used to solve cubic equations. If you have a cubic equation in the form:
x^3 + px^2 + qx + r = 0
You can use the formula to find the roots. First, find the values of 'a', 'b', and 'c' that satisfy the equation:
a + b + c = -p
ab + ac + bc = q
abc = -r
Once you have these values, you can substitute them into the (x-a)(x-b)(x-c) formula to find the roots of the cubic equation.
Example:
Let's say you have the cubic equation:
x^3 - 6x^2 + 11x - 6 = 0
Comparing this to the standard form, we get:
p = 6, q = 11, r = 6
Now, we need to find the values of 'a', 'b', and 'c' that satisfy the conditions:
a + b + c = -6
ab + ac + bc = 11
abc = 6
By trial and error, we find that the values a = 1, b = 2, and c = 3 satisfy these conditions.
Therefore, the roots of the cubic equation are:
x = 1, x = 2, x = 3
This confirms that the original cubic equation can be factored as:
(x-1)(x-2)(x-3) = 0
Conclusion
The (x-a)(x-b)(x-c) formula is a powerful tool for understanding and manipulating cubic polynomials. It provides a direct link between the roots of a cubic equation and its coefficients, making it valuable for factorization, solving equations, and gaining insights into the behavior of cubic functions.