(x%2Bb)(x%2Bc)-formula.jpeg "(x+a)(x+b)(x+c) Formula")
Expanding (x + a)(x + b)(x + c)
The expression (x + a)(x + b)(x + c) represents the product of three binomials. Expanding this expression involves multiplying each term in one binomial with each term in the other two. This process can be achieved using the following steps:
Step 1: Expand the first two binomials
First, we expand the product of the first two binomials: (x + a)(x + b). This can be done using the FOIL method (First, Outer, Inner, Last):
(x + a)(x + b) = x² + bx + ax + ab
Simplifying the expression, we get:
(x + a)(x + b) = x² + (a + b)x + ab
Step 2: Multiply the result by the third binomial
Now, we multiply the simplified result from step 1 by the third binomial (x + c):
(x² + (a + b)x + ab)(x + c)
Again, we distribute each term in the first expression by the second:
x²(x + c) + (a + b)x(x + c) + ab(x + c)
Expanding further:
x³ + cx² + (a + b)x² + (a + b)cx + abx + abc
Step 3: Combine like terms
Finally, we combine like terms to obtain the final expanded form:
x³ + (a + b + c)x² + (ab + ac + bc)x + abc
Summary
The expanded form of (x + a)(x + b)(x + c) is:
x³ + (a + b + c)x² + (ab + ac + bc)x + abc
This formula is useful for various applications in algebra and calculus, including finding the roots of a polynomial equation and analyzing the behavior of functions.