3.jpeg "(4n−3)3")
Exploring the Expression (4n-3)³
The expression (4n-3)³ represents the cube of the binomial (4n-3). This means that the expression is multiplied by itself three times:
(4n - 3)³ = (4n - 3)(4n - 3)(4n - 3)
To expand this expression, we can use the distributive property (often referred to as FOIL) repeatedly. Let's break down the process:
Step 1: Expand the first two factors
- (4n - 3)(4n - 3) = 16n² - 12n - 12n + 9
- = 16n² - 24n + 9
Step 2: Multiply the result by the third factor
- (16n² - 24n + 9)(4n - 3)
To perform this multiplication, we can distribute each term of the first factor to each term of the second factor:
- 16n² (4n - 3) = 64n³ - 48n²
- -24n (4n - 3) = -96n² + 72n
- 9 (4n - 3) = 36n - 27
Step 3: Combine like terms
Finally, add all the resulting terms together:
- 64n³ - 48n² - 96n² + 72n + 36n - 27
- = 64n³ - 144n² + 108n - 27
Therefore, the expanded form of (4n-3)³ is 64n³ - 144n² + 108n - 27.
Understanding the Result
The expanded form of (4n-3)³ is a polynomial with four terms, each with a different power of n. This type of expression is known as a cubic polynomial, due to the highest power of n being 3.
Key points to remember:
- The coefficients of the expanded form: The coefficients of the terms (64, -144, 108, -27) are determined by the specific values in the original binomial (4 and -3) and the exponent (3).
- The pattern of the powers: The exponents of n decrease from 3 to 0. This is a typical pattern for expanded binomials raised to a power.
Understanding how to expand expressions like (4n-3)³ is crucial in various mathematical contexts, including solving equations, graphing functions, and working with polynomial identities.