Question

1. use the expression to add like terms and rewrite your expression using the distributive property?

8c + 14 + 16c + 18
a. 24c + 32; 2(12c + 16)
b. 8c + 32; 8(c + 4)
c. 24c + 32; 4(6c + 8)
d. 24c + 32; 8(3c + 4)

Explanation:

Step1: Combine like terms

First, combine the \( c \)-terms and the constant terms in the expression \( 8c + 14 + 16c + 18 \).
For the \( c \)-terms: \( 8c + 16c = (8 + 16)c = 24c \)
For the constant terms: \( 14 + 18 = 32 \)
So the expression becomes \( 24c + 32 \).

Step2: Factor using distributive property

Now, we need to factor \( 24c + 32 \) using the distributive property \( ab + ac = a(b + c) \).
First, find the greatest common factor (GCF) of 24 and 32.
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
The factors of 32 are: 1, 2, 4, 8, 16, 32
The GCF of 24 and 32 is 8.
Now, factor out 8 from \( 24c + 32 \):
\( 24c + 32 = 8(3c) + 8(4) = 8(3c + 4) \) (using the distributive property \( a(b + c)=ab + ac \) in reverse)

Let's check the options:

• Option a: \( 2(12c + 16) \), GCF here is 2, but we saw GCF is 8, so incorrect.
• Option b: \( 8(c + 4) = 8c + 32 \), but we have \( 24c + 32 \), so incorrect.
• Option c: \( 4(6c + 8)=24c + 32 \), but factoring with GCF 8 is better (and the question says "using the distributive property" to rewrite, and 8 is the GCF), so incorrect.
• Option d: \( 24c + 32; 8(3c + 4) \), which matches our steps.

Answer:

d. \( 24c + 32; 8(3c + 4) \)