Question

# A biologist has found that the number of branches on a certain rare tree can be modeled by the polynomial b(y) = 4y² + y where y is the number of year

Polynomial division

A biologist has found that the number of branches on a certain rare tree can be modeled by the polynomial $$b(y) = 4y^2 + y$$ where y is the number of years after the tree reaches a height of 6 feet. The number of leaves on each branch can be modeled by the polynomial $$l(y) = 2y^3 + 3y^2 + y$$. Write a polynomial describing the total number of leaves on the tree.

2021-02-03
The polynomial describing the total number of leaves on the tree is the product of the two models. This is like multiplying the number of branches by the number of trees. If t(y)t(y) represents the total number of leaves on the tree, we write:
$$\displaystyle{t}{\left({y}\right)}={b}{\left({y}\right)}⋅{l}{\left({y}\right)}$$
$$\displaystyle{t}{\left({y}\right)}={\left({4}{y}^{{2}}+{y}\right)}{\left({2}{y}^{{3}}+{3}{y}^{{2}}+{y}\right)}$$
Use distributive property:
$$\displaystyle{t}{\left({y}\right)}={2}{y}^{{3}}{\left({4}{y}^{{2}}+{y}\right)}+{3}{y}^{{2}}{\left({4}{y}^{{2}}+{y}\right)}+{y}{\left({4}{y}^{{2}}+{y}\right)}$$
$$\displaystyle{t}{\left({y}\right)}={\left({8}{y}^{{5}}+{2}{y}^{{4}}\right)}+{\left({12}{y}^{{4}}+{3}{y}^{{3}}\right)}+{\left({4}{y}^{{3}}+{y}^{{2}}\right)}$$
$$\displaystyle{t}{\left({y}\right)}={8}{y}^{{5}}+{14}{y}^{{4}}+{7}{y}{3}+{y}^{{2}}$$