Use the chart to multiply the binomial by the trinomial.
[tex](2 y+3)(3 y^2+4 y+5)[/tex]
Which polynomial is the correct product?
A. [tex]15 y^3+17 y^2+22 y+15[/tex]
B. [tex]6 y^3+17 y^2+22 y+15[/tex]
C. [tex]6 y^3+20 y^2+22 y+15[/tex]
D. [tex]6 y^3+17 y^2+22 y+25[/tex]
Answer (1)
Multiply each term of the binomial by each term of the trinomial: ( 2 y + 3 ) ( 3 y 2 + 4 y + 5 ) = 2 y ( 3 y 2 + 4 y + 5 ) + 3 ( 3 y 2 + 4 y + 5 ) .
Distribute the terms: 2 y ( 3 y 2 + 4 y + 5 ) = 6 y 3 + 8 y 2 + 10 y and 3 ( 3 y 2 + 4 y + 5 ) = 9 y 2 + 12 y + 15 .
Add the resulting expressions: ( 6 y 3 + 8 y 2 + 10 y ) + ( 9 y 2 + 12 y + 15 ) = 6 y 3 + ( 8 y 2 + 9 y 2 ) + ( 10 y + 12 y ) + 15 .
Combine like terms to get the final answer: 6 y 3 + 17 y 2 + 22 y + 15 .
Explanation
Understanding the Problem We need to multiply the binomial ( 2 y + 3 ) by the trinomial ( 3 y 2 + 4 y + 5 ) . This means each term in the binomial must be multiplied by each term in the trinomial. We will then combine like terms to simplify the expression.
Expanding the Expression Let's multiply each term of the binomial by each term of the trinomial:
( 2 y + 3 ) ( 3 y 2 + 4 y + 5 ) = 2 y ( 3 y 2 + 4 y + 5 ) + 3 ( 3 y 2 + 4 y + 5 )
Distributing the Terms Now, distribute 2 y and 3 across the trinomial:
2 y ( 3 y 2 + 4 y + 5 ) = 6 y 3 + 8 y 2 + 10 y
3 ( 3 y 2 + 4 y + 5 ) = 9 y 2 + 12 y + 15
Combining the Expressions Next, we add the two resulting expressions:
( 6 y 3 + 8 y 2 + 10 y ) + ( 9 y 2 + 12 y + 15 ) = 6 y 3 + ( 8 y 2 + 9 y 2 ) + ( 10 y + 12 y ) + 15
Simplifying the Expression Now, combine like terms:
6 y 3 + ( 8 y 2 + 9 y 2 ) + ( 10 y + 12 y ) + 15 = 6 y 3 + 17 y 2 + 22 y + 15
Final Answer The correct product is 6 y 3 + 17 y 2 + 22 y + 15 .
Examples
Polynomial multiplication is used in various fields, such as engineering and computer graphics. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and stress on different parts of the structure. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create complex shapes and textures by combining simpler geometric forms.
