## Factoring and Solving the Polynomial (x+4)(x+1)(x-1)(x-1)

This article explores the polynomial (x+4)(x+1)(x-1)(x-1) and its properties, including how to factor it and find its roots.

### Factoring the Polynomial

The polynomial is already factored in its given form. However, we can simplify it by combining the repeated factors:

**(x+4)(x+1)(x-1)(x-1) = (x+4)(x+1)(x-1)²**

### Finding the Roots

To find the roots of the polynomial, we need to find the values of x that make the expression equal to zero. This occurs when any of the factors are equal to zero:

**x + 4 = 0**=> x = -4**x + 1 = 0**=> x = -1**x - 1 = 0**=> x = 1

Since the factor (x-1) appears twice, the root x=1 has a **multiplicity of 2**.

### Interpreting the Roots

The roots of a polynomial represent the x-intercepts of its graph. In this case, the polynomial has three roots:

- x = -4
- x = -1
- x = 1 (with a multiplicity of 2)

The multiplicity of the root x=1 indicates that the graph of the polynomial **touches the x-axis at x=1** instead of crossing it.

### Expanding the Polynomial

While the factored form is useful, we can also expand the polynomial to see its standard form:

**(x+4)(x+1)(x-1)² = (x² + 5x + 4)(x² - 2x + 1)**

Expanding further, we get:

**x⁴ + 3x³ - 9x² - 10x + 4**

This form of the polynomial is useful for understanding its general shape and behavior.

### Conclusion

The polynomial (x+4)(x+1)(x-1)² has three roots: x=-4, x=-1, and x=1 (with a multiplicity of 2). Its factored form is useful for finding its roots, while its expanded form shows its standard form and general behavior.