(x-1)(x-4).jpeg "(x+2)(x-1)(x-4)")
Factoring and Finding Roots of (x+2)(x-1)(x-4)
The expression (x+2)(x-1)(x-4) is already in factored form. This makes it easy to find the roots (or zeros) of the expression.
Understanding Factored Form
In factored form, each factor represents a linear expression that equals zero when the variable (x) takes on a specific value. This value is called the root or zero.
Finding the Roots
To find the roots, we set each factor equal to zero and solve for x:
- x + 2 = 0
- Subtract 2 from both sides: x = -2
- x - 1 = 0
- Add 1 to both sides: x = 1
- x - 4 = 0
- Add 4 to both sides: x = 4
Therefore, the roots of the expression (x+2)(x-1)(x-4) are x = -2, x = 1, and x = 4.
Expanding the Expression
We can expand the expression to get a polynomial form:
- Expand the first two factors:
- (x + 2)(x - 1) = x² + x - 2
- Multiply the result by the third factor:
- (x² + x - 2)(x - 4) = x³ - 3x² - 6x + 8
So, the expanded form of the expression is x³ - 3x² - 6x + 8.
Summary
- Factored Form: (x+2)(x-1)(x-4)
- Roots: x = -2, x = 1, x = 4
- Expanded Form: x³ - 3x² - 6x + 8
Understanding factored form is crucial in solving equations, graphing functions, and analyzing the behavior of polynomials.