(x%5E4%2B2x%5E3%2B4x%5E2%2B8x%2B16).jpeg "(x-2)(x^4+2x^3+4x^2+8x+16)")
Factoring and Expanding: (x-2)(x^4+2x^3+4x^2+8x+16)
This expression represents a multiplication of two terms: a linear binomial (x-2) and a quartic polynomial (x^4+2x^3+4x^2+8x+16). Let's explore how to simplify and understand this expression.
Expanding the Expression
To expand this expression, we can use the distributive property (also known as FOIL for binomials). This means multiplying each term in the first factor by each term in the second factor:
(x-2)(x^4+2x^3+4x^2+8x+16) =
x(x^4+2x^3+4x^2+8x+16) - 2(x^4+2x^3+4x^2+8x+16)
Expanding further, we get:
= x^5 + 2x^4 + 4x^3 + 8x^2 + 16x - 2x^4 - 4x^3 - 8x^2 - 16x - 32
Combining like terms, we arrive at the simplified form:
x^5 - 32
Understanding the Structure
The quartic polynomial (x^4+2x^3+4x^2+8x+16) has a special structure. It resembles the expansion of a sum of cubes:
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
If we let a = x and b = 2, we can see the pattern in our quartic polynomial:
x^4 + 2x^3 + 4x^2 + 8x + 16 = x^4 + 3(x^2)(2) + 3(x)(2^2) + 2^4
This pattern arises from the special case of a "sum of cubes" factorization:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In our original expression, we have the first factor (a + b) = (x + 2), and the second factor is the expanded form of (a^2 - ab + b^2). This connection reveals that our original expression is designed to be factored into a simpler form.
Factoring the Expression
Since we know the pattern, we can rewrite the expression as:
(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16) = (x - 2)(x^3 + 2^3)
Using the sum of cubes formula, we can further factor this:
= (x - 2)(x + 2)(x^2 - 2x + 4)
This gives us the fully factored form of the expression.
Conclusion
By understanding the pattern of the quartic polynomial, we can easily factor the expression (x-2)(x^4+2x^3+4x^2+8x+16) into a simpler form. This process involves recognizing the structure of the expression, utilizing the sum of cubes formula, and performing some basic algebraic manipulations.