%5E3-expand-and-simplify.jpeg "(x-2)^3 Expand And Simplify")
Expanding and Simplifying (x-2)^3
The expression (x-2)^3 represents the cube of the binomial (x-2). To expand and simplify this expression, we can use the following methods:
Method 1: Using the Binomial Theorem
The Binomial Theorem provides a formula for expanding any power of a binomial:
(a + b)^n = <sup>n</sup>C<sub>0</sub>a<sup>n</sup>b<sup>0</sup> + <sup>n</sup>C<sub>1</sub>a<sup>n-1</sup>b<sup>1</sup> + <sup>n</sup>C<sub>2</sub>a<sup>n-2</sup>b<sup>2</sup> + ... + <sup>n</sup>C<sub>n</sub>a<sup>0</sup>b<sup>n</sup>
where <sup>n</sup>C<sub>r</sub> is the binomial coefficient, calculated as:
<sup>n</sup>C<sub>r</sub> = n! / (r! * (n-r)!)
Applying this to (x-2)^3:
(x - 2)^3 = <sup>3</sup>C<sub>0</sub>x<sup>3</sup>(-2)<sup>0</sup> + <sup>3</sup>C<sub>1</sub>x<sup>2</sup>(-2)<sup>1</sup> + <sup>3</sup>C<sub>2</sub>x<sup>1</sup>(-2)<sup>2</sup> + <sup>3</sup>C<sub>3</sub>x<sup>0</sup>(-2)<sup>3</sup>
Calculating the binomial coefficients:
- <sup>3</sup>C<sub>0</sub> = 3! / (0! * 3!) = 1
- <sup>3</sup>C<sub>1</sub> = 3! / (1! * 2!) = 3
- <sup>3</sup>C<sub>2</sub> = 3! / (2! * 1!) = 3
- <sup>3</sup>C<sub>3</sub> = 3! / (3! * 0!) = 1
Substituting the coefficients and simplifying:
(x - 2)^3 = 1 * x<sup>3</sup> * 1 + 3 * x<sup>2</sup> * (-2) + 3 * x * 4 + 1 * 1 * (-8) (x - 2)^3 = x<sup>3</sup> - 6x<sup>2</sup> + 12x - 8
Method 2: Expanding by Multiplication
We can also expand (x-2)^3 by multiplying the binomial three times:
(x-2)^3 = (x-2) * (x-2) * (x-2)
First, multiply the first two binomials:
(x-2) * (x-2) = x<sup>2</sup> - 4x + 4
Then, multiply the result by the remaining (x-2):
(x<sup>2</sup> - 4x + 4) * (x-2) = x<sup>3</sup> - 6x<sup>2</sup> + 12x - 8
Result
Both methods lead to the same simplified expression:
(x-2)^3 = x<sup>3</sup> - 6x<sup>2</sup> + 12x - 8
This is the expanded and simplified form of the expression (x-2)^3.