Use the chart to multiply the binomial by the trinomial.
[tex](y+3)(y^2-3y+9)[/tex]
What is the product?
A. [tex]y^3+27[/tex]
B. [tex]y^3-27[/tex]
C. [tex]y^3-6y^2+27[/tex]
D. [tex]y^3+6y^2+27[/tex]
Answer (2)
Distribute the binomial ( y + 3 ) over the trinomial ( y 2 − 3 y + 9 ) .
Multiply each term: y ( y 2 − 3 y + 9 ) + 3 ( y 2 − 3 y + 9 ) = y 3 − 3 y 2 + 9 y + 3 y 2 − 9 y + 27 .
Combine like terms: y 3 + ( − 3 y 2 + 3 y 2 ) + ( 9 y − 9 y ) + 27 .
Simplify to get the final product: y 3 + 27 .
Explanation
Understanding the Problem We are asked to multiply the binomial ( y + 3 ) by the trinomial ( y 2 − 3 y + 9 ) . This can be done using the distributive property. We need to multiply each term in the binomial by each term in the trinomial.
Applying the Distributive Property We will use the distributive property to expand the expression: ( y + 3 ) ( y 2 − 3 y + 9 ) = y ( y 2 − 3 y + 9 ) + 3 ( y 2 − 3 y + 9 )
Expanding the Expression Now, we multiply y by each term in the trinomial: y ( y 2 − 3 y + 9 ) = y 3 − 3 y 2 + 9 y And we multiply 3 by each term in the trinomial: 3 ( y 2 − 3 y + 9 ) = 3 y 2 − 9 y + 27
Combining the Terms Now we add the two resulting expressions: ( y 3 − 3 y 2 + 9 y ) + ( 3 y 2 − 9 y + 27 ) = y 3 − 3 y 2 + 9 y + 3 y 2 − 9 y + 27
Simplifying the Expression Finally, we combine like terms: y 3 + ( − 3 y 2 + 3 y 2 ) + ( 9 y − 9 y ) + 27 = y 3 + 0 y 2 + 0 y + 27 = y 3 + 27 So the product is y 3 + 27 .
Final Answer The product of ( y + 3 ) ( y 2 − 3 y + 9 ) is y 3 + 27 .
Examples
Understanding polynomial multiplication is crucial in various fields, such as engineering and physics, where complex systems are modeled using polynomial equations. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution and stress on different parts of the structure. Multiplying polynomials helps them analyze how different design parameters affect the overall stability and safety of the bridge. Similarly, in physics, polynomial equations are used to describe the motion of objects, and multiplying polynomials can help predict the trajectory and behavior of these objects under different conditions. This skill is also fundamental in computer graphics for rendering complex shapes and textures.
To multiply the binomial ( y + 3 ) by the trinomial ( y 2 − 3 y + 9 ) , we distribute each term and simplify. The final product is y 3 + 27 , which corresponds to option A. Therefore, the answer is A : y 3 + 27 .
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