Question
Which expressions are completely factored?
Select each correct answer.
Responses
24a4+18=6(4a4+3)
24 a begin power 4 end power plus 18 equals 6 left parenthesis 4 a begin power 4 end power plus 3 right parenthesis
12a3+8a=4(3a3+2a)
12 a cubed plus 8 a equals 4 left parenthesis 3 a cubed plus 2 a right parenthesis
16a5−20a3=4a3(4a2−5)
16 a begin power 5 end power minus 20 a cubed equals 4 a cubed left parenthesis 4 a squared minus 5 right parenthesis
30a6−24a2=3a2(10a4−8)
Select each correct answer.
Responses
24a4+18=6(4a4+3)
24 a begin power 4 end power plus 18 equals 6 left parenthesis 4 a begin power 4 end power plus 3 right parenthesis
12a3+8a=4(3a3+2a)
12 a cubed plus 8 a equals 4 left parenthesis 3 a cubed plus 2 a right parenthesis
16a5−20a3=4a3(4a2−5)
16 a begin power 5 end power minus 20 a cubed equals 4 a cubed left parenthesis 4 a squared minus 5 right parenthesis
30a6−24a2=3a2(10a4−8)
Answers
Answered by
GPT-4o mini
To determine if the expressions are completely factored, we need to check if they cannot be factored further.
1. **24a⁴ + 18 = 6(4a⁴ + 3)**
- The expression inside the parentheses (4a⁴ + 3) cannot be factored further. Thus, this factorization is complete.
- **Complete**
2. **12a³ + 8a = 4(3a³ + 2a)**
- The expression inside the parentheses (3a³ + 2a) can be factored further: 3a³ + 2a = a(3a² + 2). Thus, this factorization is not complete.
- **Not Complete**
3. **16a⁵ - 20a³ = 4a³(4a² - 5)**
- The expression inside the parentheses (4a² - 5) cannot be factored further, as it does not have rational roots and cannot be expressed in simpler terms. Thus, this factorization is complete.
- **Complete**
4. **30a⁶ - 24a² = 3a²(10a⁴ - 8)**
- The expression inside the parentheses (10a⁴ - 8) can be factored as 2(5a⁴ - 4). The term (5a⁴ - 4) does not factor easily either. Since it can be expressed in terms of a difference of squares (5a⁴ - 4 = (√5 a²)² - (2)² = (√5 a² + 2)(√5 a² - 2)), this factorization is not complete.
- **Not Complete**
In summary, the completely factored expressions are:
- **24a⁴ + 18 = 6(4a⁴ + 3)**
- **16a⁵ − 20a³ = 4a³(4a² − 5)**
1. **24a⁴ + 18 = 6(4a⁴ + 3)**
- The expression inside the parentheses (4a⁴ + 3) cannot be factored further. Thus, this factorization is complete.
- **Complete**
2. **12a³ + 8a = 4(3a³ + 2a)**
- The expression inside the parentheses (3a³ + 2a) can be factored further: 3a³ + 2a = a(3a² + 2). Thus, this factorization is not complete.
- **Not Complete**
3. **16a⁵ - 20a³ = 4a³(4a² - 5)**
- The expression inside the parentheses (4a² - 5) cannot be factored further, as it does not have rational roots and cannot be expressed in simpler terms. Thus, this factorization is complete.
- **Complete**
4. **30a⁶ - 24a² = 3a²(10a⁴ - 8)**
- The expression inside the parentheses (10a⁴ - 8) can be factored as 2(5a⁴ - 4). The term (5a⁴ - 4) does not factor easily either. Since it can be expressed in terms of a difference of squares (5a⁴ - 4 = (√5 a²)² - (2)² = (√5 a² + 2)(√5 a² - 2)), this factorization is not complete.
- **Not Complete**
In summary, the completely factored expressions are:
- **24a⁴ + 18 = 6(4a⁴ + 3)**
- **16a⁵ − 20a³ = 4a³(4a² − 5)**