(a) Calculate the value of $u^{3}-2 u^{2} v+u v^{2}-3 u v+3 v^{2}$ if $u-v=3$ . $\newline$ (b) Find the smallest possible value of $3 a^{2}+27 b^{2}+5 c^{2}-18 a b-30 c+125$ .

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Algebra 2

Evaluate expression when a complex numbers and a variable term is given

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Q. (a) Calculate the value of

u^{3}-2 u^{2} v+u v^{2}-3 u v+3 v^{2}

u-v=3

\newline

(b) Find the smallest possible value of

3 a^{2}+27 b^{2}+5 c^{2}-18 a b-30 c+125

Find $u$ in terms of $v$ : Given $u-v=3$ , let's find $u$ in terms of $v$ : $u = v + 3$ .
Substitute and simplify expression: Substitute $u = v + 3$ into the expression $u^3 - 2u^2v + uv^2 - 3uv + 3v^2$ .
Expand and simplify expression: $(u^3)$ becomes $(v + 3)^3$ .
Combine terms and simplify: $(2u^2v)$ becomes $2(v + 3)^2v$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2.$ $(3uv)$ becomes $3(v + 3)v.$
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .Now, complete the square for the $3(v + 3)v$ $7$ term: $3(v + 3)v$ $8$ , factor out a $3(v + 3)v$ $9$ to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $0$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .Now, complete the square for the $3(v + 3)v$ $7$ term: $3(v + 3)v$ $8$ , factor out a $3(v + 3)v$ $9$ to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $0$ .Complete the square by adding and subtracting $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $1$ inside the parentheses: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $2$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .Now, complete the square for the $3(v + 3)v$ $7$ term: $3(v + 3)v$ $8$ , factor out a $3(v + 3)v$ $9$ to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $0$ .Complete the square by adding and subtracting $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $1$ inside the parentheses: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $2$ .This simplifies to $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $3$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .Now, complete the square for the $3(v + 3)v$ $7$ term: $3(v + 3)v$ $8$ , factor out a $3(v + 3)v$ $9$ to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $0$ .Complete the square by adding and subtracting $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $1$ inside the parentheses: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $2$ .This simplifies to $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $3$ .Combine all terms: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $4$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .Now, complete the square for the $3(v + 3)v$ $7$ term: $3(v + 3)v$ $8$ , factor out a $3(v + 3)v$ $9$ to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $0$ .Complete the square by adding and subtracting $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $1$ inside the parentheses: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $2$ .This simplifies to $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $3$ .Combine all terms: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $4$ .Simplify to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $5$ .
Find minimum value: $(uv^2)$ becomes $(v + 3)v^2$ . $(3uv)$ becomes $3(v + 3)v$ .Now, expand and simplify the expression: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ . $(v + 3)^3 = v^3 + 9v^2 + 27v + 27$ . $2(v + 3)^2v = 2(v^2 + 6v + 9)v = 2v^3 + 12v^2 + 18v$ . $(v + 3)v^2 = v^3 + 3v^2$ . $3(v + 3)v = 3v^2 + 9v$ .Combine all the terms: $v^3 + 9v^2 + 27v + 27 - (2v^3 + 12v^2 + 18v) + v^3 + 3v^2 - (3v^2 + 9v) + 3v^2$ .Simplify the expression: $(v + 3)v^2$ $0$ .Combine like terms: $(v + 3)v^2$ $1$ .The terms simplify to: $(v + 3)v^2$ $2$ cancels out, and we're left with $(v + 3)v^2$ $3$ .The $(v + 3)v^2$ $4$ terms: $(v + 3)v^2$ $5$ simplify to $(v + 3)v^2$ $6$ .The $(v + 3)v^2$ $7$ terms: $(v + 3)v^2$ $8$ simplify to $(v + 3)v^2$ $9$ .The constant term is just $(3uv)$ $0$ .So, the simplified expression is $(3uv)$ $0$ .Now, let's find the smallest possible value of $(3uv)$ $2$ .To find the minimum, we can complete the square for the $(3uv)$ $3$ and $(3uv)$ $4$ terms.For the $(3uv)$ $3$ terms: $(3uv)$ $6$ , factor out a $(3uv)$ $7$ to get $(3uv)$ $8$ .Complete the square by adding and subtracting $(3uv)$ $9$ inside the parentheses: $3(v + 3)v$ $0$ .This simplifies to $3(v + 3)v$ $1$ .Now, for the $(3uv)$ $4$ terms: $3(v + 3)v$ $3$ , add the $3(v + 3)v$ $4$ from the previous step to get $3(v + 3)v$ $5$ .Combine the terms: $3(v + 3)v$ $6$ .Now, complete the square for the $3(v + 3)v$ $7$ term: $3(v + 3)v$ $8$ , factor out a $3(v + 3)v$ $9$ to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $0$ .Complete the square by adding and subtracting $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $1$ inside the parentheses: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $2$ .This simplifies to $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $3$ .Combine all terms: $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $4$ .Simplify to get $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $5$ .The smallest value occurs when the squared terms are zero, so the minimum value is $(v + 3)^3 - 2(v + 3)^2v + (v + 3)v^2 - 3(v + 3)v + 3v^2$ $6$ .

(a) Calculate the value of u3−2u2v+uv2−3uv+3v2 u^{3}-2 u^{2} v+u v^{2}-3 u v+3 v^{2} u3−2u2v+uv2−3uv+3v2 if u−v=3 u-v=3 u−v=3.\newline(b) Find the smallest possible value of 3a2+27b2+5c2−18ab−30c+125 3 a^{2}+27 b^{2}+5 c^{2}-18 a b-30 c+125 3a2+27b2+5c2−18ab−30c+125.

(a) Calculate the value of $u^{3}-2 u^{2} v+u v^{2}-3 u v+3 v^{2}$ if $u-v=3$ . $\newline$ (b) Find the smallest possible value of $3 a^{2}+27 b^{2}+5 c^{2}-18 a b-30 c+125$ .