(x%2B2)(x%2B3)(x%2B6).jpeg "(x-1)(x+2)(x+3)(x+6)")
Factoring and Solving the Polynomial (x-1)(x+2)(x+3)(x+6)
The expression (x-1)(x+2)(x+3)(x+6) represents a polynomial in factored form. Let's explore what this means and how to use it.
Understanding the Factored Form
This form tells us directly the roots or zeros of the polynomial. A root is a value of x that makes the polynomial equal to zero.
- The Zero Product Property: The key to understanding factored polynomials is the zero product property: If the product of several factors is zero, then at least one of the factors must be zero.
Applying this to our polynomial:
- (x-1) = 0 => x = 1
- (x+2) = 0 => x = -2
- (x+3) = 0 => x = -3
- (x+6) = 0 => x = -6
Therefore, the roots of the polynomial are x = 1, x = -2, x = -3, and x = -6.
Expanding the Polynomial
While the factored form is useful for finding roots, sometimes we need to see the polynomial in its standard form. We can do this by expanding the expression:
-
Start with the first two factors: (x-1)(x+2) = x² + x - 2
-
Multiply the result by the third factor: (x² + x - 2)(x+3) = x³ + 4x² + x - 6
-
Finally, multiply by the last factor: (x³ + 4x² + x - 6)(x+6) = x⁴ + 10x³ + 21x² - 2x - 36
Therefore, the expanded form of the polynomial is x⁴ + 10x³ + 21x² - 2x - 36.
Applications
Understanding factored and expanded forms of polynomials is crucial in many areas of mathematics:
- Solving equations: By setting the polynomial equal to zero, we can find its roots, which are solutions to the equation.
- Graphing functions: Knowing the roots helps us locate the x-intercepts of the graph of the polynomial function.
- Calculus: Factored forms can be useful for calculating derivatives and integrals of polynomials.
In conclusion, the expression (x-1)(x+2)(x+3)(x+6) provides a compact representation of a polynomial with specific roots. Understanding the factored and expanded forms of this polynomial allows us to solve equations, analyze graphs, and explore more advanced mathematical concepts.