(x%2B1)(x%2B3)(x%2B5)%2B7.jpeg "(x-1)(x+1)(x+3)(x+5)+7")
Factoring and Exploring the Polynomial (x-1)(x+1)(x+3)(x+5)+7
This article delves into the intriguing polynomial expression (x-1)(x+1)(x+3)(x+5)+7 and explores its factorization and some interesting properties.
Expanding the Expression
First, let's expand the expression by multiplying the factors:
(x-1)(x+1)(x+3)(x+5)+7 = (x² - 1)(x² + 8x + 15) + 7
Expanding further:
= x⁴ + 8x³ + 14x² + 8x - 8 + 7
= x⁴ + 8x³ + 14x² + 8x - 1
Factoring the Expression
While the expression appears complex, it's not easily factorable using traditional methods. However, we can use a clever trick to simplify it.
Let's consider the expression:
x² + 4x + 3 = (x+1)(x+3)
Now, notice that our original expression can be rewritten as:
(x-1)(x+1)(x+3)(x+5)+7 = + 7
Now, we can substitute (x² + 4x + 3) with the factored form:
= [(x+1)(x+3) - 4] [(x+1)(x+3) + 2] + 7
= (x+1)²(x+3)² - 2(x+1)(x+3) + 1
This expression looks much simpler and can be factored using the quadratic formula:
Let y = (x+1)(x+3)
Then the expression becomes:
y² - 2y + 1 = (y-1)²
Substituting back:
= [(x+1)(x+3) - 1]²
= (x² + 4x + 2)²
Properties and Analysis
Now that we've factored the expression, we can analyze some of its properties:
- Roots: The polynomial has a double root at x = -2 ± √2. This means the graph touches the x-axis at these points but does not cross it.
- Symmetry: The graph of the polynomial is symmetric about the line x = -2. This is because the expression is a perfect square.
- Minimum Value: The polynomial has a minimum value at x = -2, which is 0. This is because the square of any real number is always non-negative.
Conclusion
The expression (x-1)(x+1)(x+3)(x+5)+7, although initially appearing complex, can be simplified and factored into a perfect square. This reveals its inherent symmetry and minimum value, highlighting the power of algebraic manipulation and the elegance of mathematical expressions.