(x%2B3)(x-4).jpeg "(x+1)(x+3)(x-4)")
Exploring the Polynomial (x+1)(x+3)(x-4)
This article will explore the polynomial (x+1)(x+3)(x-4) and its properties. We will delve into its expansion, roots, and how to analyze its behavior.
Expansion
To begin, let's expand the polynomial:
(x+1)(x+3)(x-4) = (x^2 + 4x + 3)(x - 4)
= x^3 - 4x^2 + 4x^2 - 16x + 3x - 12
= x^3 - 13x - 12
Now we have the polynomial in its standard form, which makes it easier to work with.
Roots
The roots of a polynomial are the values of x that make the polynomial equal to zero. To find the roots of (x+1)(x+3)(x-4), we can set each factor equal to zero:
- x + 1 = 0 => x = -1
- x + 3 = 0 => x = -3
- x - 4 = 0 => x = 4
Therefore, the roots of the polynomial are -1, -3, and 4.
Behavior
The behavior of a polynomial can be determined by its leading coefficient and its degree. In this case, the leading coefficient is 1 (positive) and the degree is 3 (odd).
- Leading Coefficient: A positive leading coefficient means the graph will rise to the right.
- Degree: An odd degree means the graph will fall to the left and rise to the right.
Based on these characteristics, we can conclude that the graph of the polynomial will have the following features:
- It will rise to the right as x approaches positive infinity.
- It will fall to the left as x approaches negative infinity.
- It will cross the x-axis at the roots -1, -3, and 4.
Summary
The polynomial (x+1)(x+3)(x-4) expands to x^3 - 13x - 12. Its roots are -1, -3, and 4. The graph of this polynomial will rise to the right, fall to the left, and cross the x-axis at the points -1, -3, and 4.