Let P be the set of polynomials. Let a, b, c, and d be elements of P such that b and d are non-zero elements of P. Which of the following is true regarding the sum below? A. The sum is a polynomial. B. The sum is a rational expression. C. The sum is an integer. D. The sum is a rational number.

8 and 10

Step-by-step explanation:

Given

+ =

Combine the fractions on the left side

+ =

= ( cross- multiply )

40(2x + 2) = 9x(x + 2) ← distribute parenthesis on both sides

80x + 80 = 9x² + 18x ← subtract 80x + 80 from both sides

0 = 9x² -62x - 80 ← in standard form

To factorise the quadratic

Consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term

product = 9 × - 80 = - 720 and sum = - 62

Factors are - 72 and + 10

Use these factors to split the x- term

9x² - 72x + 10x - 80 = 0 ( factor the first/second and third/fourth terms )

9x(x - 8) + 10(x - 8) = 0 ← factor out (x - 8) from each term

(x - 8)(9x + 10) = 0

Equate each factor to zero and solve for x

x - 8 = 0 ⇒ x = 8

9x + 10 = 0 ⇒ 9x = - 10 ⇒ x = -

However, x is an integer ⇒ x = 8

The integers are 8 and 8 + 2 = 10