(x%2Bb)(x%2Bc)-formula-expansion.jpeg "(x+a)(x+b)(x+c) Formula Expansion")
Expanding (x+a)(x+b)(x+c)
The expression (x+a)(x+b)(x+c) represents the product of three linear binomials. Expanding this expression means multiplying all the terms together to get a polynomial expression. Here's a step-by-step guide on how to expand it and the resulting formula:
Step 1: Expand the first two binomials
Begin by expanding the first two binomials, (x+a)(x+b) using the FOIL method (First, Outer, Inner, Last):
(x+a)(x+b) = x² + ax + bx + ab
Step 2: Multiply the result by the third binomial
Now, multiply the expanded result from step 1 by the third binomial (x+c):
(x² + ax + bx + ab)(x+c) = x³ + ax² + bx² + abx + cx² + acx + bcx + abc
Step 3: Combine like terms
Finally, combine all the like terms to get the fully expanded expression:
(x+a)(x+b)(x+c) = x³ + (a+b+c)x² + (ab+ac+bc)x + abc
The Formula
Therefore, the expanded form of (x+a)(x+b)(x+c) is:
x³ + (a+b+c)x² + (ab+ac+bc)x + abc
Applications
This formula is useful in various contexts, including:
- Algebra: It helps simplify and solve cubic equations.
- Calculus: It's used in finding the derivative and integral of cubic functions.
- Geometry: It can be applied to problems involving volumes of rectangular prisms.
Example
Let's expand (x+2)(x+3)(x+4) using the formula:
- a = 2, b = 3, c = 4
- x³ + (2+3+4)x² + (23 + 24 + 34)x + 23*4
- x³ + 9x² + 26x + 24
Therefore, the expanded form of (x+2)(x+3)(x+4) is x³ + 9x² + 26x + 24.
By understanding the expansion of (x+a)(x+b)(x+c), you gain a powerful tool for manipulating and analyzing cubic expressions.