(2x%5E2-x%2B4).jpeg "(x^2-x+1)(2x^2-x+4)")
Expanding the Expression (x^2-x+1)(2x^2-x+4)
This article will guide you through the process of expanding the expression (x^2-x+1)(2x^2-x+4). This involves multiplying two trinomials, which can seem daunting at first glance. However, by using the distributive property and combining like terms, we can simplify the expression and arrive at a polynomial in standard form.
Step 1: Applying the Distributive Property
We can expand the expression by applying the distributive property twice. First, distribute the first trinomial (x^2-x+1) over each term in the second trinomial (2x^2-x+4):
(x^2-x+1)(2x^2-x+4) = (x^2-x+1)(2x^2) + (x^2-x+1)(-x) + (x^2-x+1)(4)
Step 2: Expanding Each Term
Now, distribute each term in the first trinomial over the terms inside the parentheses:
(x^2-x+1)(2x^2) = 2x^4 - 2x^3 + 2x^2 (x^2-x+1)(-x) = -x^3 + x^2 - x (x^2-x+1)(4) = 4x^2 - 4x + 4
Step 3: Combining Like Terms
Finally, combine the like terms from each expansion to obtain the final simplified polynomial:
2x^4 - 2x^3 + 2x^2 - x^3 + x^2 - x + 4x^2 - 4x + 4 = 2x^4 - 3x^3 + 7x^2 - 5x + 4
Conclusion
Therefore, the expanded form of (x^2-x+1)(2x^2-x+4) is 2x^4 - 3x^3 + 7x^2 - 5x + 4. By following the steps outlined above, you can successfully expand any product of trinomials or polynomials, regardless of their complexity.