Question

explain 2 modeling with polynomial multiplication
read explain 2 part a and complete your turn #1 (adapted from lesson 6.2).
example 2 find the polynomial function modeling the desired relationship
a. an object thrown in the air has a velocity after t seconds that can be described by v(t) (in meters/second). the kinetic energy of the object is given by (k=\frac{1}{2}mv^{2}). if the mass is 2 kg, find an expression for the kinetic energy.
(k = \frac{1}{2}(2)(-9.8t + 24)^{2})
(k=(-9.8t + 24)^{2}) simplify
(=(-9.8t + 24)(-9.8t + 24)) write out the same expression twice
(=-9.8t(-9.8t + 24)+24(-9.8t + 24)) multiply -9.8t first then 24
(=96.04t^{2}-470.4t + 576) combine like terms

1. 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.

Explanation:

Step1: Multiply the two polynomials

$b(y)\times l(y)=(4y^{2}+y)(2y^{3}+3y^{2}+y)$

Step2: Apply distributive property

$4y^{2}(2y^{3}+3y^{2}+y)+y(2y^{3}+3y^{2}+y)$
$=(4y^{2}\times2y^{3}+4y^{2}\times3y^{2}+4y^{2}\times y)+(y\times2y^{3}+y\times3y^{2}+y\times y)$

Step3: Use exponent - rule $a^{m}\times a^{n}=a^{m + n}$

$=(8y^{5}+12y^{4}+4y^{3})+(2y^{4}+3y^{3}+y^{2})$

Step4: Combine like terms

$=8y^{5}+(12y^{4}+2y^{4})+(4y^{3}+3y^{3})+y^{2}$
$=8y^{5}+14y^{4}+7y^{3}+y^{2}$

Answer:

$8y^{5}+14y^{4}+7y^{3}+y^{2}$