(-2x%5E2%2B5x%2B3).jpeg "(4x^2-2x+1)(-2x^2+5x+3)")
Expanding the Expression: (4x^2 - 2x + 1)(-2x^2 + 5x + 3)
This article will guide you through the process of expanding the given expression: (4x^2 - 2x + 1)(-2x^2 + 5x + 3). We will utilize the distributive property (also known as the FOIL method) to achieve this.
The Distributive Property (FOIL Method)
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In the context of multiplying binomials, we can use the acronym FOIL to remember the steps:
- First: Multiply the first terms of each binomial.
- Outer: Multiply the outer terms of each binomial.
- Inner: Multiply the inner terms of each binomial.
- Last: Multiply the last terms of each binomial.
Expanding the Expression
Let's apply the FOIL method to our expression:
1. First: (4x^2)(-2x^2) = -8x^4
2. Outer: (4x^2)(5x) = 20x^3
3. Inner: (-2x)( -2x^2) = 4x^3
4. Last: (-2x)(3) = -6x
5. First: (1)(-2x^2) = -2x^2
6. Outer: (1)(5x) = 5x
7. Inner: (1)(-2x^2) = -2x^2
8. Last: (1)(3) = 3
Now, we have expanded the expression and can combine like terms:
-8x^4 + 20x^3 + 4x^3 - 6x - 2x^2 + 5x - 2x^2 + 3
Simplifying the expression, we get:
-8x^4 + 24x^3 - 4x^2 - x + 3
Conclusion
By applying the distributive property (FOIL method), we have successfully expanded the expression (4x^2 - 2x + 1)(-2x^2 + 5x + 3) into its simplified form: -8x^4 + 24x^3 - 4x^2 - x + 3.