(x+4)(x+1)(x-1)(x-1)

3 min read Jun 16, 2024

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.