(x%5E2-5x%2B25)-x(x-4)%5E2%2B16x.jpeg "(x+5)(x^2-5x+25)-x(x-4)^2+16x")
Simplifying the Expression: (x+5)(x^2-5x+25)-x(x-4)^2+16x
This article explores the process of simplifying the algebraic expression: (x+5)(x^2-5x+25)-x(x-4)^2+16x. We'll break down each step to understand how the expression simplifies.
Expanding the Expressions
Firstly, we need to expand the expressions by applying the distributive property and the appropriate formulas.
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(x+5)(x^2-5x+25): This is a special case of the sum of cubes formula: (a+b)(a^2-ab+b^2) = a^3 + b^3. Applying this, we get x^3 + 125.
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x(x-4)^2: We first expand the square term (x-4)^2 = x^2 - 8x + 16, and then multiply by x, resulting in x^3 - 8x^2 + 16x.
Combining Like Terms
Now we have the expression: x^3 + 125 - (x^3 - 8x^2 + 16x) + 16x.
We can simplify this by distributing the negative sign and combining the like terms:
x^3 + 125 - x^3 + 8x^2 - 16x + 16x
This simplifies further to: 8x^2 + 125
Final Result
Therefore, the simplified form of the expression (x+5)(x^2-5x+25)-x(x-4)^2+16x is 8x^2 + 125.