Question

factor $2t^{4}+2,000ty^{3}$ completely.

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the two terms \(2t^4\) and \(2000ty^3\). The GCF of 2 and 2000 is 2, and the GCF of \(t^4\) and \(ty^3\) is \(t\). So we factor out \(2t\) from both terms.
\(2t^4 + 2000ty^3 = 2t(t^3 + 1000y^3)\)

Step2: Apply the sum of cubes formula

Notice that \(t^3 + 1000y^3\) is a sum of cubes, since \(1000y^3=(10y)^3\). The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). Here, \(a = t\) and \(b = 10y\).
So we factor \(t^3 + (10y)^3\) using the sum of cubes formula:
\(t^3 + (10y)^3=(t + 10y)(t^2 - t(10y)+(10y)^2)=(t + 10y)(t^2 - 10ty + 100y^2)\)

Step3: Combine the factors

Substitute the factored form of \(t^3 + 1000y^3\) back into the expression from Step 1:
\(2t(t^3 + 1000y^3)=2t(t + 10y)(t^2 - 10ty + 100y^2)\)

Answer:

\(2t(t + 10y)(t^2 - 10ty + 100y^2)\)