Of course! To determine the total number of leaves on the tree, we'll be working with two given polynomials:
1. The number of branches on the tree, [tex]\( b(y) = 4y^2 + y \)[/tex]
2. The number of leaves on each branch, [tex]\( l(y) = 2y^3 + 3y^2 + y \)[/tex]
To find the total number of leaves on the tree, we need to multiply the polynomial representing the number of branches by the polynomial representing the number of leaves per branch. In other words, we need to compute:
[tex]\[ \text{Total leaves} = (4y^2 + y) \times (2y^3 + 3y^2 + y) \][/tex]
Let's break this multiplication down step-by-step:
1. Distribute [tex]\( 4y^2 \)[/tex] to each term in [tex]\( l(y) \)[/tex]:
[tex]\[ 4y^2 \times (2y^3 + 3y^2 + y) = 4y^2 \times 2y^3 + 4y^2 \times 3y^2 + 4y^2 \times y \][/tex]
This results in:
[tex]\[ 4y^2 \times 2y^3 = 8y^5 \][/tex]
[tex]\[ 4y^2 \times 3y^2 = 12y^4 \][/tex]
[tex]\[ 4y^2 \times y = 4y^3 \][/tex]
2. Next, distribute [tex]\( y \)[/tex] to each term in [tex]\( l(y) \)[/tex]:
[tex]\[ y \times (2y^3 + 3y^2 + y) = y \times 2y^3 + y \times 3y^2 + y \times y \][/tex]
This results in:
[tex]\[ y \times 2y^3 = 2y^4 \][/tex]
[tex]\[ y \times 3y^2 = 3y^3 \][/tex]
[tex]\[ y \times y = y^2 \][/tex]
3. Combine all these terms together:
[tex]\[ 8y^5 + 12y^4 + 4y^3 + 2y^4 + 3y^3 + y^2 \][/tex]
Next, we combine like terms in the resulting polynomial:
- The [tex]\( y^5 \)[/tex] term is: [tex]\( 8y^5 \)[/tex]
- Combine the [tex]\( y^4 \)[/tex] terms: [tex]\( 12y^4 + 2y^4 = 14y^4 \)[/tex]
- Combine the [tex]\( y^3 \)[/tex] terms: [tex]\( 4y^3 + 3y^3 = 7y^3 \)[/tex]
- The [tex]\( y^2 \)[/tex] term is: [tex]\( y^2 \)[/tex]
So, the final polynomial describing the total number of leaves on the tree is:
[tex]\[ 8y^5 + 14y^4 + 7y^3 + y^2 \][/tex]
This polynomial represents the total number of leaves on the tree as a function of [tex]\( y \)[/tex].
