Answer:
D. P + Q
Step-by-step explanation:
Given:
P = 8y⁴ + 6y³ + 8y
Q = (5y² - 4y)(3y² + 7)
Required:
Determine which operation will give an expression equivalent to 23y⁴ - 6y³ + 35y² - 20y.
SOLUTION:
Perform each operation given to see which of them gives an expression equivalent to 23y⁴ - 6y³ + 35y² - 20y.
Q - P:
P = 8y⁴ + 6y³ + 8y
Q = (5y² - 4y)(3y² + 7)
Q - P = (5y² - 4y)(3y² + 7) - (8y⁴ + 6y³ + 8y)
(5y²(3y² + 7) -4y(3y² + 7)) - (8y⁴ + 6y³ + 8y)
(15y⁴ + 35y² - 12y³ - 28y) - (8y⁴ + 6y³ + 8y)
Open parentheses
15y⁴ + 35y² - 12y³ - 28y - 8y⁴ - 6y³ - 8y
Collect like terms
15y⁴ - 8y⁴ - 12y³ - 6y³ + 35y² - 28y - 8y
7y⁴ - 18y³ + 35y² - 36y
Therefore, P - Q does not give us an expression equivalent to 23y⁴ - 6y³ + 35y² - 20y.
PQ:
P = 8y⁴ + 6y³ + 8y
Q = (5y² - 4y)(3y² + 7) = (15y⁴ + 35y² - 12y³ - 28y)
PQ = (8y⁴ + 6y³ + 8y)(15y⁴ + 35y² - 12y³ - 28y)
=(8y⁴(15y⁴ + 35y² - 12y³ - 28y) +6y³(8y⁴ + 6y³ + 8y)(15y⁴ + 35y² - 12y³ - 28y) +8y(8y⁴ + 6y³ + 8y)(15y⁴ + 35y² - 12y³ - 28y))
= 120y⁸ . . . . .
Note: You don't need to perform this operation further anymore. It would definitely not give us the equivalent expression we are looking for. The degree of the leading term (120y⁸) is way too greater than the degree of the leading term (23y⁴) of the equivalent expression we are looking for.
Thus, PQ cannot give us an expression equivalent to 23y⁴ - 6y³ + 35y² - 20y.
P - Q:
P = 8y⁴ + 6y³ + 8y
Q = (5y² - 4y)(3y² + 7) = (15y⁴ + 35y² - 12y³ - 28y)
P - Q = (8y⁴ + 6y³ + 8y) - (15y⁴ + 35y² - 12y³ - 28y)
Open parentheses
8y⁴ + 6y³ + 8y - 15y⁴ - 35y² + 12y³ + 28y
Collect like terms
8y⁴ - 15y⁴ + 6y³ + 12y³ - 35y² + 8y + 28y
-7y⁴ + 18y³ - 35y² + 36y
Therefore, P - Q cannot give us an expression equivalent to 23y⁴ - 6y³ + 35y² - 20y.
P + Q:
P = 8y⁴ + 6y³ + 8y
Q = (5y² - 4y)(3y² + 7) = (15y⁴ + 35y² - 12y³ - 28y)
P + Q = (8y⁴ + 6y³ + 8y) + (15y⁴ + 35y² - 12y³ - 28y)
Open parentheses
8y⁴ + 6y³ + 8y + 15y⁴ + 35y² - 12y³ - 28y
Collect like terms
8y⁴ + 15y⁴ + 6y³ - 12y³ + 35y² + 8y - 28y
23y³ - 6y³ + 35y² - 20y
Therefore, P + Q will result in an expression equivalent to 23y⁴ - 6y³ + 35y² - 20y.
The answer is D.