Select the correct answer.
Consider polynomials [tex]$P$[/tex] and [tex]$Q$[/tex].
[tex]
\begin{array}{l}
P=8 y^4+6 y^3+8 y \\
Q=\left(5 y^2-4 y\right)\left(3 y^2+7\right)
\end{array}
[/tex]
Which operation results in an expression equivalent to [tex]$23 y^4-6 y^3+35 y^2-20 y$[/tex]?
A. [tex]$P-Q$[/tex]
B. [tex]$Q-P$[/tex]
C. [tex]$P+Q$[/tex]
D. [tex]$P Q$[/tex]
Answer (2)
Expand polynomial Q = ( 5 y 2 − 4 y ) ( 3 y 2 + 7 ) to get Q = 15 y 4 − 12 y 3 + 35 y 2 − 28 y .
Calculate P + Q = ( 8 y 4 + 6 y 3 + 8 y ) + ( 15 y 4 − 12 y 3 + 35 y 2 − 28 y ) .
Simplify P + Q to get 23 y 4 − 6 y 3 + 35 y 2 − 20 y .
The operation that results in the target expression is P + Q , so the answer is P + Q .
Explanation
Problem Analysis We are given two polynomials, P = 8 y 4 + 6 y 3 + 8 y and Q = ( 5 y 2 − 4 y ) ( 3 y 2 + 7 ) . Our goal is to find which operation, P − Q , Q − P , P + Q , or PQ , results in the expression 23 y 4 − 6 y 3 + 35 y 2 − 20 y .
Expanding Polynomial Q First, we need to expand the polynomial Q . We have
Q = ( 5 y 2 − 4 y ) ( 3 y 2 + 7 ) = 5 y 2 ( 3 y 2 + 7 ) − 4 y ( 3 y 2 + 7 ) = 15 y 4 + 35 y 2 − 12 y 3 − 28 y = 15 y 4 − 12 y 3 + 35 y 2 − 28 y .
Calculating P + Q Now, let's compute P + Q :
P + Q = ( 8 y 4 + 6 y 3 + 8 y ) + ( 15 y 4 − 12 y 3 + 35 y 2 − 28 y ) = ( 8 + 15 ) y 4 + ( 6 − 12 ) y 3 + 35 y 2 + ( 8 − 28 ) y = 23 y 4 − 6 y 3 + 35 y 2 − 20 y .
Identifying the Correct Operation We see that P + Q = 23 y 4 − 6 y 3 + 35 y 2 − 20 y , which is the target expression.
Final Answer Therefore, the correct answer is C. P + Q .
Examples
Polynomials are used in various fields such as engineering, physics, and economics. For example, in engineering, polynomials can be used to model the trajectory of a projectile or the stress on a bridge. In economics, polynomials can be used to model cost and revenue functions. Understanding polynomial operations like addition, subtraction, and multiplication is crucial for solving real-world problems in these fields. For instance, if P ( x ) represents the cost of producing x items and Q ( x ) represents the revenue from selling x items, then Q ( x ) − P ( x ) represents the profit.
The operation that results in the expression 23 y 4 − 6 y 3 + 35 y 2 − 20 y is P + Q . After expanding polynomial Q and simplifying P + Q , we confirmed that it gives the target expression. Therefore, the correct answer is option C: P + Q .
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