Question
The area of a rectangle is given. Factor out the GCF to find expressions for the missing dimensions.
A=15a^3b^5+21a^2b^4+12a^3b^2
L=?
W=?
A=15a^3b^5+21a^2b^4+12a^3b^2
L=?
W=?
Answers
GPT-4o mini
To factor out the GCF (Greatest Common Factor) from the expression for the area \( A = 15a^3b^5 + 21a^2b^4 + 12a^3b^2 \), we first need to identify the GCF of the coefficients and the variables in each term.
1. **Coefficients**: The coefficients are 15, 21, and 12.
- The GCF of 15, 21, and 12 is 3.
2. **Variable \( a \)**:
- The terms contain \( a^3 \), \( a^2 \), and \( a^3 \).
- The GCF for the variable \( a \) is \( a^2 \) (the lowest power of \( a \)).
3. **Variable \( b \)**:
- The terms contain \( b^5 \), \( b^4 \), and \( b^2 \).
- The GCF for the variable \( b \) is \( b^2 \) (the lowest power of \( b \)).
Now, we can combine these to find the overall GCF of the entire expression:
\[
\text{GCF} = 3a^2b^2
\]
Next, we will factor \( 3a^2b^2 \) out of each term:
\[
A = 3a^2b^2 \left( \frac{15a^3b^5}{3a^2b^2} + \frac{21a^2b^4}{3a^2b^2} + \frac{12a^3b^2}{3a^2b^2} \right)
\]
Calculating the terms inside the parentheses:
1. For the first term:
\[
\frac{15a^3b^5}{3a^2b^2} = 5a^{3-2}b^{5-2} = 5ab^3
\]
2. For the second term:
\[
\frac{21a^2b^4}{3a^2b^2} = 7b^{4-2} = 7b^2
\]
3. For the third term:
\[
\frac{12a^3b^2}{3a^2b^2} = 4a^{3-2} = 4a
\]
Putting it all together, we have:
\[
A = 3a^2b^2(5ab^3 + 7b^2 + 4a)
\]
To make this more interpretable, we can rename \( A \) as \( (L)(W) \), where \( L \) and \( W \) represent the length and width of the rectangle, respectively. Thus, we can express:
\[
L = 3a^2b^2 \quad \text{and} \quad W = 5ab^3 + 7b^2 + 4a
\]
So the missing dimensions can be represented as:
\[
L = 3a^2b^2
\]
\[
W = 5ab^3 + 7b^2 + 4a
\]
1. **Coefficients**: The coefficients are 15, 21, and 12.
- The GCF of 15, 21, and 12 is 3.
2. **Variable \( a \)**:
- The terms contain \( a^3 \), \( a^2 \), and \( a^3 \).
- The GCF for the variable \( a \) is \( a^2 \) (the lowest power of \( a \)).
3. **Variable \( b \)**:
- The terms contain \( b^5 \), \( b^4 \), and \( b^2 \).
- The GCF for the variable \( b \) is \( b^2 \) (the lowest power of \( b \)).
Now, we can combine these to find the overall GCF of the entire expression:
\[
\text{GCF} = 3a^2b^2
\]
Next, we will factor \( 3a^2b^2 \) out of each term:
\[
A = 3a^2b^2 \left( \frac{15a^3b^5}{3a^2b^2} + \frac{21a^2b^4}{3a^2b^2} + \frac{12a^3b^2}{3a^2b^2} \right)
\]
Calculating the terms inside the parentheses:
1. For the first term:
\[
\frac{15a^3b^5}{3a^2b^2} = 5a^{3-2}b^{5-2} = 5ab^3
\]
2. For the second term:
\[
\frac{21a^2b^4}{3a^2b^2} = 7b^{4-2} = 7b^2
\]
3. For the third term:
\[
\frac{12a^3b^2}{3a^2b^2} = 4a^{3-2} = 4a
\]
Putting it all together, we have:
\[
A = 3a^2b^2(5ab^3 + 7b^2 + 4a)
\]
To make this more interpretable, we can rename \( A \) as \( (L)(W) \), where \( L \) and \( W \) represent the length and width of the rectangle, respectively. Thus, we can express:
\[
L = 3a^2b^2 \quad \text{and} \quad W = 5ab^3 + 7b^2 + 4a
\]
So the missing dimensions can be represented as:
\[
L = 3a^2b^2
\]
\[
W = 5ab^3 + 7b^2 + 4a
\]