The correct option is B.
What is the distributive property of multiplication over addition?
For numbers, a, b, and c distributivity of multiplication over addition says
a(b + c) = ab + bc
Given
We have two algebraic expressions; a² - 1, a + 1.
Add the expressions
Let their sum be g(a). On adding the two expressions
g(a) = (a² - 1) + (a + 1).
Using commutativity of addition of numbers,
g(a) = (a² - 1) + (1 + a).
= ((a² - 1) + 1) + a (Associative property)
= ((a² + (-1)) + 1) + a (a - b = a + (-b) )
= (a² + ((-1) + 1)) + a (Associative property)
= (a² + 0) + a (a + (-a) = 0)
= a² + a (a + 0 = a)
= a(a + 1) (Distributive property: a(b + c) = ab + bc)
⇒ (a² - 1) + (a + 1) = a(a + 1).
Option B matches the above result.
The correct option is B.
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Question
which of the following is equivalent to the sum of the expressions a² - 1 and a + 1?
(A) a(a - 1)
(B) a(a + 1)
(C) a²(a + 1)
(D) a²(a - 1)
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