Question

The following sequence of steps shows why you can square binomial the short way. For each step, name the definition or property that justifies the step.

(a+b)²
a. = (a+b)(a+b)
b. = (a +b)(a) + (a+b)(b)
c. = a² + ba + ab + b²
d. = a² + 2ab + b²

Answer 1

a. definition of exponent

b. distributive property

c. distributive property

d. combining like terms

Explanation:

You want the properties that justify the steps on expanding the square of a binomial.

a. = (a+b)(a+b)

The factor is repeated a number of times equal to the exponent. This is the definition of an exponent.

b. = (a +b)(a) + (a+b)(b)

The first factor multiplies each term of the second factor. This is an example of the distributive property.

c. = a² + ba + ab + b²

The second factor multiplies each term of the first factor. This is an example of the distributive property.

d. = a² + 2ab + b²

Terms ba and ab are combined. This "collecting terms" essentially makes use of the commutative property of multiplication and the distributive property:

ba +ab = ab +ab . . . . . commutative property of multiplication

ab +ab = (1 +1)ab . . . . . distributive property

= 2ab . . . . . . . properties of integers