Question

use the chart to multiply the binomial by the trinomial.
(2y + 3)(3y² + 4y + 5)
chart with empty cells
which polynomial is the correct product?
○ 15y³ + 17y² + 22y + 15
○ 6y³ + 17y² + 22y + 15
○ 6y³ + 20y² + 22y + 15
○ 6y³ + 17y² + 22y + 25

Explanation:

Step1: Apply Distributive Property (FOIL-like for binomial-trinomial)

Multiply each term in the binomial \((2y + 3)\) by each term in the trinomial \((3y^2 + 4y + 5)\).

• Multiply \(2y\) by \(3y^2\): \(2y \times 3y^2 = 6y^3\)
• Multiply \(2y\) by \(4y\): \(2y \times 4y = 8y^2\)
• Multiply \(2y\) by \(5\): \(2y \times 5 = 10y\)
• Multiply \(3\) by \(3y^2\): \(3 \times 3y^2 = 9y^2\)
• Multiply \(3\) by \(4y\): \(3 \times 4y = 12y\)
• Multiply \(3\) by \(5\): \(3 \times 5 = 15\)

Step2: Combine Like Terms

• For \(y^3\): \(6y^3\) (only term)
• For \(y^2\): \(8y^2 + 9y^2 = 17y^2\)
• For \(y\): \(10y + 12y = 22y\)
• Constant term: \(15\)

Combine all terms: \(6y^3 + 17y^2 + 22y + 15\)

Answer:

\(6y^3 + 17y^2 + 22y + 15\) (corresponding to the second option, e.g., B. \(6y^3 + 17y^2 + 22y + 15\))