Question
Find the common factor of all of the terms of the polynomial below.
14x2 – 12x
Group of answer choices
2x2
4x2
2x
4x
Flag question: Question 2
Question 21 pts
Factor the polynomial.
2x4 + 4x3 + 6x2
Group of answer choices
2x2(x2 + 2x + 3)
x2(2x3 + 3x2 + 2x)
2x3(x + 2) + 6x2
(2x2 + 3x)(x2 + 2x)
Flag question: Question 3
Question 31 pts
Factor the polynomial: 2x(x – 4) + 7(x – 4)
Group of answer choices
(x – 4)(2x + 7)
(2x – 4)(x + 7)
14x(x – 4)
(x – 4)(2x – 7)
Flag question: Question 4
Question 41 pts
Use the grouping method to factor the polynomial below completely.
x3 + 2x2 + 3x + 6
Group of answer choices
(x2 + 3)(x + 3)
(x2 + 2)(x + 3)
(x2 + 2)(x + 2)
(x2 + 3)(x + 2)
Flag question: Question 5
Question 51 pts
Use the grouping method to factor 4x3 + 20x2 – 3x – 15.
Group of answer choices
(x – 5)(4x2 + 3)
(x2 – 5)(4x + 3)
2x(x + 1)(2x – 15)
(x + 5)(4x2 – 3)
14x2 – 12x
Group of answer choices
2x2
4x2
2x
4x
Flag question: Question 2
Question 21 pts
Factor the polynomial.
2x4 + 4x3 + 6x2
Group of answer choices
2x2(x2 + 2x + 3)
x2(2x3 + 3x2 + 2x)
2x3(x + 2) + 6x2
(2x2 + 3x)(x2 + 2x)
Flag question: Question 3
Question 31 pts
Factor the polynomial: 2x(x – 4) + 7(x – 4)
Group of answer choices
(x – 4)(2x + 7)
(2x – 4)(x + 7)
14x(x – 4)
(x – 4)(2x – 7)
Flag question: Question 4
Question 41 pts
Use the grouping method to factor the polynomial below completely.
x3 + 2x2 + 3x + 6
Group of answer choices
(x2 + 3)(x + 3)
(x2 + 2)(x + 3)
(x2 + 2)(x + 2)
(x2 + 3)(x + 2)
Flag question: Question 5
Question 51 pts
Use the grouping method to factor 4x3 + 20x2 – 3x – 15.
Group of answer choices
(x – 5)(4x2 + 3)
(x2 – 5)(4x + 3)
2x(x + 1)(2x – 15)
(x + 5)(4x2 – 3)
Answers
GPT-4o mini
Let's analyze each question one by one:
### Question 1: Find the common factor of all the terms of the polynomial
**Polynomial:**
\[ 14x^2 - 12x \]
**Common Factor:**
- The coefficients are 14 and -12. The greatest common factor (GCF) of 14 and 12 is 2.
- The variable part: Both terms have \( x \), with the lowest power being \( x \).
Thus, the common factor of the polynomial is:
\[ 2x \]
**Answer: `2x`**
---
### Question 2: Factor the polynomial
**Polynomial:**
\[ 2x^4 + 4x^3 + 6x^2 \]
**Factoring out the GCF:**
- The GCF of the coefficients is 2.
- The lowest power of \( x \) is \( x^2 \).
Thus, factoring out \( 2x^2 \):
\[ 2x^2(x^2 + 2x + 3) \]
**Answer: `2x^2(x^2 + 2x + 3)`**
---
### Question 3: Factor the polynomial
**Polynomial:**
\[ 2x(x - 4) + 7(x - 4) \]
**Factoring out the common factor:**
The common factor is \( (x - 4) \):
\[ (x - 4)(2x + 7) \]
**Answer: `(x - 4)(2x + 7)`**
---
### Question 4: Use the grouping method to factor the polynomial below completely.
**Polynomial:**
\[ x^3 + 2x^2 + 3x + 6 \]
**Grouping:**
Group the terms:
\[ (x^3 + 2x^2) + (3x + 6) \]
- Factor out \( x^2 \) from the first group and \( 3 \) from the second group:
\[ x^2(x + 2) + 3(x + 2) \]
- Now factor out \( (x + 2) \):
\[ (x + 2)(x^2 + 3) \]
None of the answer choices match, but let's check the answer choices closely to see any close matching.
- \( (x^2 + 2)(x + 3) \) is not valid since it does not fit.
In a standard context, if `x^2 + 3` does not factor further, the answer could be approximated by recognizing that multiple choices provided have possible issues with presentation. But one of the option answers seems to indicate mismatched content.
**Answer: The best noted from provided should be checked with regard to visual misalignment**.
- Formulate a correct answer traceable back toward `(x + 2)(x^2 + 3)`.
---
### Question 5: Use the grouping method to factor 4x^3 + 20x^2 - 3x - 15.
**Polynomial:**
\[ 4x^3 + 20x^2 - 3x - 15 \]
**Grouping:**
Group the terms:
\[ (4x^3 + 20x^2) + (-3x - 15) \]
- Factor out the GCF from both groups:
\[ 4x^2(x + 5) - 3(x + 5) \]
- Now factor out \( (x + 5) \):
\[ (x + 5)(4x^2 - 3) \]
Verifying the answer choices aligns with:
**Answer: `(x + 5)(4x^2 - 3)`**
---
The answers provide feedback on framing and ensure the checking of responsive format in all adaptive processing for consistent understanding is key for assessment correctness!
### Question 1: Find the common factor of all the terms of the polynomial
**Polynomial:**
\[ 14x^2 - 12x \]
**Common Factor:**
- The coefficients are 14 and -12. The greatest common factor (GCF) of 14 and 12 is 2.
- The variable part: Both terms have \( x \), with the lowest power being \( x \).
Thus, the common factor of the polynomial is:
\[ 2x \]
**Answer: `2x`**
---
### Question 2: Factor the polynomial
**Polynomial:**
\[ 2x^4 + 4x^3 + 6x^2 \]
**Factoring out the GCF:**
- The GCF of the coefficients is 2.
- The lowest power of \( x \) is \( x^2 \).
Thus, factoring out \( 2x^2 \):
\[ 2x^2(x^2 + 2x + 3) \]
**Answer: `2x^2(x^2 + 2x + 3)`**
---
### Question 3: Factor the polynomial
**Polynomial:**
\[ 2x(x - 4) + 7(x - 4) \]
**Factoring out the common factor:**
The common factor is \( (x - 4) \):
\[ (x - 4)(2x + 7) \]
**Answer: `(x - 4)(2x + 7)`**
---
### Question 4: Use the grouping method to factor the polynomial below completely.
**Polynomial:**
\[ x^3 + 2x^2 + 3x + 6 \]
**Grouping:**
Group the terms:
\[ (x^3 + 2x^2) + (3x + 6) \]
- Factor out \( x^2 \) from the first group and \( 3 \) from the second group:
\[ x^2(x + 2) + 3(x + 2) \]
- Now factor out \( (x + 2) \):
\[ (x + 2)(x^2 + 3) \]
None of the answer choices match, but let's check the answer choices closely to see any close matching.
- \( (x^2 + 2)(x + 3) \) is not valid since it does not fit.
In a standard context, if `x^2 + 3` does not factor further, the answer could be approximated by recognizing that multiple choices provided have possible issues with presentation. But one of the option answers seems to indicate mismatched content.
**Answer: The best noted from provided should be checked with regard to visual misalignment**.
- Formulate a correct answer traceable back toward `(x + 2)(x^2 + 3)`.
---
### Question 5: Use the grouping method to factor 4x^3 + 20x^2 - 3x - 15.
**Polynomial:**
\[ 4x^3 + 20x^2 - 3x - 15 \]
**Grouping:**
Group the terms:
\[ (4x^3 + 20x^2) + (-3x - 15) \]
- Factor out the GCF from both groups:
\[ 4x^2(x + 5) - 3(x + 5) \]
- Now factor out \( (x + 5) \):
\[ (x + 5)(4x^2 - 3) \]
Verifying the answer choices aligns with:
**Answer: `(x + 5)(4x^2 - 3)`**
---
The answers provide feedback on framing and ensure the checking of responsive format in all adaptive processing for consistent understanding is key for assessment correctness!