%5E4.jpeg "(x-7)^4")
Exploring the Expansion of (x-7)^4
The expression (x-7)^4 represents the fourth power of the binomial (x-7). This means we are multiplying (x-7) by itself four times:
(x-7)^4 = (x-7) * (x-7) * (x-7) * (x-7)
While we could directly multiply these terms, it's a tedious process. Instead, we can use the Binomial Theorem to simplify the expansion.
The Binomial Theorem
The Binomial Theorem provides a formula for expanding any power of a binomial:
(a + b)^n = a^n + nCa^(n-1)b + nC2a^(n-2)b^2 + ... + nCn-1ab^(n-1) + b^n
Where:
- n is the power of the binomial
- nCk represents the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying the Binomial Theorem to (x-7)^4
Let's apply the Binomial Theorem to expand (x-7)^4:
- a = x
- b = -7
- n = 4
Therefore:
(x-7)^4 = x^4 + 4Cx^3(-7) + 4C2x^2(-7)^2 + 4C3x(-7)^3 + (-7)^4
Now, we need to calculate the binomial coefficients:
- 4C0 = 1
- 4C1 = 4
- 4C2 = 6
- 4C3 = 4
- 4C4 = 1
Substituting the coefficients and simplifying:
(x-7)^4 = x^4 - 28x^3 + 294x^2 - 1372x + 2401
Understanding the Expansion
The expansion of (x-7)^4 results in a polynomial with five terms:
- x^4: The highest power term, representing x multiplied by itself four times.
- -28x^3: This term is derived from the product of x^3 and -7, multiplied by the binomial coefficient 4C1.
- 294x^2: This term arises from the product of x^2 and (-7)^2, multiplied by the binomial coefficient 4C2.
- -1372x: This term is the product of x and (-7)^3, multiplied by the binomial coefficient 4C3.
- 2401: This is the constant term, representing (-7)^4.
Conclusion
The expansion of (x-7)^4 provides us with a polynomial expression that represents the fourth power of the binomial. Understanding the Binomial Theorem allows us to efficiently expand such expressions without lengthy multiplication. This is particularly useful in algebra and calculus, where manipulating and understanding binomial expansions is crucial for solving various problems.