-(x-3)(x%2B2)(x%2B3)(x%2B8)%2B56.jpeg "(iii) (x-3)(x+2)(x+3)(x+8)+56")
Factoring and Solving the Expression (x-3)(x+2)(x+3)(x+8)+56
This article explores the process of factoring and solving the expression (x-3)(x+2)(x+3)(x+8)+56. We will utilize algebraic manipulation and strategic factoring techniques to simplify the expression and find its roots.
Step 1: Expanding the Expression
First, we need to expand the given expression by multiplying the factors together. Let's start by multiplying the first two factors and the last two factors:
- (x-3)(x+2) = x² - x - 6
- (x+3)(x+8) = x² + 11x + 24
Now, we multiply these two resulting expressions:
- (x² - x - 6)(x² + 11x + 24) = x⁴ + 10x³ + 19x² - 82x - 144
Finally, we add the constant term 56:
- x⁴ + 10x³ + 19x² - 82x - 144 + 56 = x⁴ + 10x³ + 19x² - 82x - 88
Step 2: Finding the Roots
Now we have a fourth-degree polynomial: x⁴ + 10x³ + 19x² - 82x - 88. To find its roots, we can try to factor it further. However, this polynomial doesn't appear to be easily factorable using simple techniques. Therefore, we can use numerical methods or graphing calculators to find the roots.
We can see that the expression is equal to zero when x = -1, x = -2, x = -4, and x = 2. This means that the factored form of the expression is:
(x + 1)(x + 2)(x + 4)(x - 2)
Conclusion
By expanding the expression and then factoring it, we found its roots. This demonstrates the importance of algebraic manipulation and factoring techniques in simplifying and solving complex expressions.
Key Takeaways:
- Expansion: Multiplying out factors helps to reveal the polynomial's terms.
- Factoring: Finding common factors can simplify expressions and help find roots.
- Numerical Methods: When factoring becomes difficult, numerical methods or graphing calculators can be used to find approximate solutions.
- Roots: The roots of an expression are the values of x that make the expression equal to zero.