(x%2B2)(x%2B3)(x%2B4)(x%2B5)-simplify.jpeg "(x+1)(x+2)(x+3)(x+4)(x+5) Simplify")
Simplifying the Product of Polynomials: $(x+1)(x+2)(x+3)(x+4)(x+5)$
When dealing with the product of multiple polynomials, simplifying the expression can be a daunting task. However, with the right approach and techniques, it can be broken down into manageable parts. In this article, we will explore how to simplify the product of polynomials: $(x+1)(x+2)(x+3)(x+4)(x+5)$.
Step 1: Expand the Product
To begin, we need to expand the product of polynomials using the distributive property of multiplication over addition. We will start by multiplying the first two polynomials:
$(x+1)(x+2) = x^2 + 3x + 2$
Next, we will multiply the result by the third polynomial:
$(x^2 + 3x + 2)(x+3) = x^3 + 6x^2 + 11x + 6$
We will continue this process, multiplying the result by the fourth and fifth polynomials:
$(x^3 + 6x^2 + 11x + 6)(x+4) = x^4 + 10x^3 + 35x^2 + 50x + 24$
$(x^4 + 10x^3 + 35x^2 + 50x + 24)(x+5) = x^5 + 15x^4 + 85x^3 + 225x^2 + 274x + 120$
Step 2: Simplify the Expression
After expanding the product, we can simplify the expression by combining like terms:
$x^5 + 15x^4 + 85x^3 + 225x^2 + 274x + 120$
This is the simplified form of the product of polynomials: $(x+1)(x+2)(x+3)(x+4)(x+5)$. Note that the resulting polynomial has a degree of 5, which is the sum of the degrees of the original polynomials.
Conclusion
Simplifying the product of polynomials requires patience and attention to detail. By following the steps outlined above, we can break down the expression into manageable parts and arrive at the simplified form. The final result is a polynomial of degree 5, which can be used for further calculations or graphical analysis.
I hope this article has helped you in simplifying the product of polynomials. If you have any more questions or need further assistance, feel free to ask!
