Question

Justifying Multiplication of Linear Expressions
A 2-column table with 5 rows. First column is labeled Step with entries one-fourth x (5 x minus 1), (one-fourth x) (5 x) + (one-fourth x)(negative 1), one-fourth times 5 times x times x + (negative 1) times one-fourth times x, (one-fourth times 5) (x times x) + (negative 1 times one-fourth) x, 5 over 4 x squared minus one-fourth x. Second column is labeled Reason with entries given, 1, 2, 3, 4.

The table shows the multiplication of two linear expressions. Select the correct reason for each step.


Reason 1:

Reason 2:

Reason 3:

Reason 4:

Answer 1

Reason 1: Distributive Property of multiplication over subtraction (a(b−c)=ab−ac).

Reason 2: Commutative property of multiplication (reorder factors) to write each product as numbers times variables: (1/4·5·x·x) + (−1·1/4·x).

Reason 3: Associative property of multiplication (regroup factors) to separate numeric coefficients and variable factors: (1/4·5)(x·x) + (−1·1/4)x.

Reason 4: Simplify the numeric products and use x·x = x^2: 5/4 x^2 − 1/4 x.