Answer:
[tex]\frac{10}{7}\cdot a^{3}\cdot y^{4} - \frac{1}{7}\cdot a^{4}\cdot y^{3} - \frac{5}{21}\cdot a^5\cdot y^{2}}[/tex]
Step-by-step explanation:
According to the statement, we have the algebraic equation[tex]\frac{2}{7}\cdot {a^{2}\cdot y^{2}}\cdot \left(5\cdot a\cdot y^{2} - \frac{1}{2}\cdot {a}^{2}\cdot y - \frac{5}{6}\cdot a^{3}\right)[/tex] and we must to rewrite it as a polynomial. Since there are two variables: [tex]a[/tex], [tex]y[/tex], we must observe the following definition of polynomial:
[tex]p(a, y) = \Sigma\limits_{i = 0}^{n} c_{i}\cdot a^{i}\cdot y^{n-i}[/tex] (1)
Where:
[tex]i[/tex] - Index.
[tex]c_{i}[/tex] - i-th Coefficient of the polynomial.
[tex]n[/tex] - Grade of the polynomial.
By means of algebraic handling, we have the following result:
1) [tex]\frac{2}{7}\cdot {a^{2}\cdot y^{2}}\cdot \left(5\cdot a\cdot y^{2} - \frac{1}{2}\cdot {a}^{2}\cdot y - \frac{5}{6}\cdot a^{3}\right)[/tex] Given
2) [tex]\left(\frac{2}{7}\cdot a^{2}\cdot y^{2} \right)\cdot \left(5\cdot a\cdot y^{2}\right) + \left(\frac{2}{7}\cdot a^{2}\cdot y^{2} \right) \cdot \left(-\frac{1}{2}\cdot a^{2}\cdot y \right) + \left(\frac{2}{7}\cdot a^{2}\cdot y^{2} \right)\cdot \left(-\frac{5}{6}\cdot a^{3}\right)[/tex] Associative and distributive properties/[tex](-a)\cdot b = -a\cdot b[/tex]
3) [tex]\frac{10}{7}\cdot a^{3}\cdot y^{4} - \frac{1}{7}\cdot a^{4}\cdot y^{3} - \frac{5}{21}\cdot a^5\cdot y^{2}}[/tex] Associative and commutative properties/[tex]\frac{a}{b}\times \frac{c}{d} = \frac{a\cdot c}{b\cdot d}[/tex]/[tex](-a)\cdot b = -a\cdot b[/tex]/Definition of subtraction/[tex]\frac{a\cdot c}{b\cdot c} = \frac{a}{b}[/tex]/Result
