Question

Write as a polynomial - \dfrac{ 2 }{ 7 } { a }^{ 2 } { y }^{ 7 } \left( 5a { y }^{ 2 } - \dfrac{ 1 }{ 2 } { a }^{ 2 } y- \dfrac{ 5 }{ 6 } { a }^{ 3 } \right)

Answer 1

`(10)/(7)\cdot a^(3)\cdot y^(4) - (1)/(7)\cdot a^(4)\cdot y^(3) - (5)/(21)\cdot a^5\cdot y^(2)}`

Explanation:

According to the statement, we have the algebraic equation
`(2)/(7)\cdot {a^(2)\cdot y^(2)}\cdot \left(5\cdot a\cdot y^(2) - (1)/(2)\cdot {a}^(2)\cdot y - (5)/(6)\cdot a^(3)\right)` and we must to rewrite it as a polynomial. Since there are two variables:
`a`,
`y`, we must observe the following definition of polynomial:

`p(a, y) = \Sigma\limits_(i = 0)^(n) c_(i)\cdot a^(i)\cdot y^(n-i)` (1)

Where:

`i` - Index.

`c_(i)` - i-th Coefficient of the polynomial.

`n` - Grade of the polynomial.

By means of algebraic handling, we have the following result:

1)
`(2)/(7)\cdot {a^(2)\cdot y^(2)}\cdot \left(5\cdot a\cdot y^(2) - (1)/(2)\cdot {a}^(2)\cdot y - (5)/(6)\cdot a^(3)\right)` Given

2)
`\left((2)/(7)\cdot a^(2)\cdot y^(2) \right)\cdot \left(5\cdot a\cdot y^(2)\right) + \left((2)/(7)\cdot a^(2)\cdot y^(2) \right) \cdot \left(-(1)/(2)\cdot a^(2)\cdot y \right) + \left((2)/(7)\cdot a^(2)\cdot y^(2) \right)\cdot \left(-(5)/(6)\cdot a^(3)\right)` Associative and distributive properties/
`(-a)\cdot b = -a\cdot b`

3)
`(10)/(7)\cdot a^(3)\cdot y^(4) - (1)/(7)\cdot a^(4)\cdot y^(3) - (5)/(21)\cdot a^5\cdot y^(2)}` Associative and commutative properties/
`(a)/(b)* (c)/(d) = (a\cdot c)/(b\cdot d)`/
`(-a)\cdot b = -a\cdot b`/Definition of subtraction/
`(a\cdot c)/(b\cdot c) = (a)/(b)`/Result