The first event requires 30x flyers, each with 6x² graphics. The second event requires 24x flyers, each with 4x graphics. Question: What is the total number of graphics needed for both events, and how can you factor it? Guide Questions: 1. What expression represents the number of graphics for each event? 2. Combine both expressions into one polynomial. 3. What are the coefficients and variables in each term? 4. What is the greatest monomial factor common to both terms? 5. Factor out the GCMF. What does the factored form tell you about how materials are grouped?
Answer (1)
To solve this problem, we'll follow a step-by-step approach to determine the total number of graphics needed for both events and factor the expression.
Expression for Each Event:
For the first event, you need 30x flyers, each with 6 x 2 graphics. The total graphics for the first event are: 30 x × 6 x 2 = 180 x 3
For the second event, you need 24x flyers, each with 4 x graphics. The total graphics for the second event are: 24 x × 4 x = 96 x 2
Combine Both Expressions: Combine both expressions to get the total number of graphics required: 180 x 3 + 96 x 2
Coefficients and Variables in Each Term:
First term: Coefficient is 180, variable is x 3 .
Second term: Coefficient is 96, variable is x 2 .
Greatest Common Monomial Factor (GCMF):
The greatest common factor of the coefficients 180 and 96 is 12.
The greatest common factor of the variables x 3 and x 2 is x 2 .
Therefore, the GCMF is 12 x 2 .
Factor the Expression: Factor out the GCMF 12 x 2 from the polynomial: 180 x 3 + 96 x 2 = 12 x 2 ( 15 x + 8 )
The factored form 12 x 2 ( 15 x + 8 ) indicates that materials are grouped such that there is a common factor of 12 x 2 across both terms in the polynomial, simplifying distribution and possibly storage or production logistics. This grouping helps us see the shared basis between the events and unify the resources needed for both.
