(2m%5E2-3mn%2B8n%5E2).jpeg "(4m+n)(2m^2-3mn+8n^2)")
Multiplying Binomials: (4m + n)(2m² - 3mn + 8n²)
This article will explore the multiplication of the binomials (4m + n) and (2m² - 3mn + 8n²). We will use the distributive property to expand the product and simplify the resulting expression.
Understanding the Distributive Property
The distributive property states that for any numbers a, b, and c:
a(b + c) = ab + ac
This property allows us to multiply a sum by a number by distributing the multiplication to each term inside the parentheses.
Expanding the Product
To expand (4m + n)(2m² - 3mn + 8n²), we need to distribute each term in the first binomial to every term in the second binomial.
Step 1: Distribute 4m.
4m(2m² - 3mn + 8n²) = 8m³ - 12m²n + 32mn²
Step 2: Distribute n.
n(2m² - 3mn + 8n²) = 2m²n - 3mn² + 8n³
Step 3: Combine the results of Step 1 and Step 2.
(4m + n)(2m² - 3mn + 8n²) = (8m³ - 12m²n + 32mn²) + (2m²n - 3mn² + 8n³)
Step 4: Simplify by combining like terms.
(4m + n)(2m² - 3mn + 8n²) = **8m³ - 10m²n + 29mn² + 8n³**
Conclusion
Therefore, the product of (4m + n) and (2m² - 3mn + 8n²) is 8m³ - 10m²n + 29mn² + 8n³. By applying the distributive property and combining like terms, we successfully expanded the product and obtained a simplified expression.