Question
A binomial has been partially factored, as shown below:
x2 − 25 = (x + 5)(_____)
Which of the following binomials represents the missing factor?
(x − 5)
(x + 5)
(x − 20)
(x + 20)
Question 2(Multiple Choice Worth 4 points)
(07.02 MC)
Rewrite 6x2 + 2xy + 9x + 3y in factored form.
(2x + 3)(3x + y)
(2x + y)(3x + 3)
(3x + 2)(3x + y)
(3x + y)(2x + 1)
Question 3(Multiple Choice Worth 4 points)
(07.04 MC)
Factor completely x3 + 6x2 − 9x − 54.
(x + 3)(x − 3)(x + 6)
(x + 3)(x − 3)(x − 6)
(x2 + 9)(x + 6)
(x2 − 9)(x + 6)
Question 4(Multiple Choice Worth 4 points)
(07.02 MC)
Identify the factors of 6ab − 8a + 21b − 28.
(2a + 4)(3b − 7)
(2a − 4)(3b + 7)
(2a + 7)(3b − 4)
(2a − 7)(3b + 4)
Question 5(Multiple Choice Worth 4 points)
(07.04 MC)
Factor completely 81x4 − 16.
(3x − 2)(3x − 2)(9x2 + 4)
(3x − 2)(3x + 2)(9x2 − 4)
(3x − 2)(3x + 2)(9x2 + 4)
(3x + 2)(3x + 2)(9x2 + 4)
Question 6(Multiple Choice Worth 4 points)
(07.03 LC)
Determine which polynomial is a perfect square trinomial.
25x2 − 40x − 16
9a2 − 20a − 25
25b2 − 15b + 9
16x2 − 56x + 49
Question 7(Multiple Choice Worth 4 points)
(07.01, 07.02 MC)
Factor the greatest common factor: 28a3b4 + 20a2b2 − 16ab3.
4ab(7a2b + 5a − 4b2)
4ab2(7a2b + 5a − 4b)
4ab2(7a2b2 + 5a − 4b)
4ab(7a2b3 + 5a − 4b)
Question 8(Multiple Choice Worth 4 points)
(07.01, 07.02 LC)
Factor the greatest common factor: −4y2 + 12y − 16.
−1(4y2 − 12y + 16)
−4y(y2 − 3y + 4)
−4(y2 − 3y + 4)
−4(y2 + 3y − 4)
Question 9(Multiple Choice Worth 4 points)
(07.02 MC)
Factor completely 2x3 + 14x2 + 6x + 42.
2[(x2 + 3)(x + 7)]
(2x2 + 6)(x + 7)
(x2 + 3)(2x + 14)
2(x3 + 7x2 + 3x + 21)
Question 10(Multiple Choice Worth 4 points)
(07.04 MC)
Identify the factors of x2 + 25y2.
(x + 5y)(x + 5y)
(x + 5y)(x − 5y)
(x − 5y)(x − 5y)
Prime
Question 11(Multiple Choice Worth 4 points)
(07.03 MC)
Factor completely 2x2 − 6x − 36.
2(x − 3)(x + 6)
2(x − 6)(x + 3)
(2x − 4)(x + 9)
(2x − 9)(x + 4)
Question 12(Multiple Choice Worth 4 points)
(07.03 MC)
Sam decides to build a square garden. If the area of the garden is 4x2 + 28x + 49 square feet, what is the length of one side of the garden?
(2x + 7) feet
(7x + 2) feet
(2x − 7) feet
(7x − 2) feet
Question 13(Multiple Choice Worth 4 points)
(07.03 LC)
Determine the factors of x2 − 7x − 10.
(x + 2)(x − 5)
Prime
(x − 2)(x + 5)
(x + 10)(x − 1)
Question 14(Multiple Choice Worth 4 points)
(07.03 MC)
What are the factors of 4x2 + 23x − 6?
(4x − 6)(x + 1)
(4x − 1)(x + 6)
(4x − 2)(x + 3)
(4x − 3)(x + 2)
Question 15(Multiple Choice Worth 4 points)
(07.03 MC)
Factor completely 3x2 − 30x + 63.
3(x − 7)(x − 3)
3(x − 7)(x + 3)
3(x − 3)(x + 7)
3(x + 3)(x + 7)
x2 − 25 = (x + 5)(_____)
Which of the following binomials represents the missing factor?
(x − 5)
(x + 5)
(x − 20)
(x + 20)
Question 2(Multiple Choice Worth 4 points)
(07.02 MC)
Rewrite 6x2 + 2xy + 9x + 3y in factored form.
(2x + 3)(3x + y)
(2x + y)(3x + 3)
(3x + 2)(3x + y)
(3x + y)(2x + 1)
Question 3(Multiple Choice Worth 4 points)
(07.04 MC)
Factor completely x3 + 6x2 − 9x − 54.
(x + 3)(x − 3)(x + 6)
(x + 3)(x − 3)(x − 6)
(x2 + 9)(x + 6)
(x2 − 9)(x + 6)
Question 4(Multiple Choice Worth 4 points)
(07.02 MC)
Identify the factors of 6ab − 8a + 21b − 28.
(2a + 4)(3b − 7)
(2a − 4)(3b + 7)
(2a + 7)(3b − 4)
(2a − 7)(3b + 4)
Question 5(Multiple Choice Worth 4 points)
(07.04 MC)
Factor completely 81x4 − 16.
(3x − 2)(3x − 2)(9x2 + 4)
(3x − 2)(3x + 2)(9x2 − 4)
(3x − 2)(3x + 2)(9x2 + 4)
(3x + 2)(3x + 2)(9x2 + 4)
Question 6(Multiple Choice Worth 4 points)
(07.03 LC)
Determine which polynomial is a perfect square trinomial.
25x2 − 40x − 16
9a2 − 20a − 25
25b2 − 15b + 9
16x2 − 56x + 49
Question 7(Multiple Choice Worth 4 points)
(07.01, 07.02 MC)
Factor the greatest common factor: 28a3b4 + 20a2b2 − 16ab3.
4ab(7a2b + 5a − 4b2)
4ab2(7a2b + 5a − 4b)
4ab2(7a2b2 + 5a − 4b)
4ab(7a2b3 + 5a − 4b)
Question 8(Multiple Choice Worth 4 points)
(07.01, 07.02 LC)
Factor the greatest common factor: −4y2 + 12y − 16.
−1(4y2 − 12y + 16)
−4y(y2 − 3y + 4)
−4(y2 − 3y + 4)
−4(y2 + 3y − 4)
Question 9(Multiple Choice Worth 4 points)
(07.02 MC)
Factor completely 2x3 + 14x2 + 6x + 42.
2[(x2 + 3)(x + 7)]
(2x2 + 6)(x + 7)
(x2 + 3)(2x + 14)
2(x3 + 7x2 + 3x + 21)
Question 10(Multiple Choice Worth 4 points)
(07.04 MC)
Identify the factors of x2 + 25y2.
(x + 5y)(x + 5y)
(x + 5y)(x − 5y)
(x − 5y)(x − 5y)
Prime
Question 11(Multiple Choice Worth 4 points)
(07.03 MC)
Factor completely 2x2 − 6x − 36.
2(x − 3)(x + 6)
2(x − 6)(x + 3)
(2x − 4)(x + 9)
(2x − 9)(x + 4)
Question 12(Multiple Choice Worth 4 points)
(07.03 MC)
Sam decides to build a square garden. If the area of the garden is 4x2 + 28x + 49 square feet, what is the length of one side of the garden?
(2x + 7) feet
(7x + 2) feet
(2x − 7) feet
(7x − 2) feet
Question 13(Multiple Choice Worth 4 points)
(07.03 LC)
Determine the factors of x2 − 7x − 10.
(x + 2)(x − 5)
Prime
(x − 2)(x + 5)
(x + 10)(x − 1)
Question 14(Multiple Choice Worth 4 points)
(07.03 MC)
What are the factors of 4x2 + 23x − 6?
(4x − 6)(x + 1)
(4x − 1)(x + 6)
(4x − 2)(x + 3)
(4x − 3)(x + 2)
Question 15(Multiple Choice Worth 4 points)
(07.03 MC)
Factor completely 3x2 − 30x + 63.
3(x − 7)(x − 3)
3(x − 7)(x + 3)
3(x − 3)(x + 7)
3(x + 3)(x + 7)
Answers
GPT-4o mini
Here's the complete solution for the questions presented:
**Question 1:**
The remaining factor that completes the equation \(x^2 - 25 = (x + 5)(\_\_\_\_) \) is the difference of squares:
1. \(x^2 - 25 = (x + 5)(x - 5)\)
**Answer:** (x − 5)
---
**Question 2:**
To factor \(6x^2 + 2xy + 9x + 3y\), we can group terms:
- Factor by grouping: \(2x(3x + y) + 3(3x + y) \)
- Result: \((2x + 3)(3x + y)\)
**Answer:** (2x + 3)(3x + y)
---
**Question 3:**
To factor \(x^3 + 6x^2 - 9x - 54\):
1. Grouping: \((x^3 + 6x^2) + (-9x - 54)\)
2. Factor out: \(x^2(x + 6) - 9(x + 6)\)
3. Result: \((x + 6)(x^2 - 9)\)
4. \(x^2 - 9\) is a difference of squares so: \((x + 6)(x + 3)(x - 3)\)
**Answer:** (x + 3)(x − 3)(x + 6)
---
**Question 4:**
To factor \(6ab - 8a + 21b - 28\):
1. Grouping: \((6ab - 8a) + (21b - 28)\)
2. Factor out: \(2a(3b - 4) + 7(3b - 4)\)
3. Result: \((2a + 7)(3b - 4)\)
**Answer:** (2a + 7)(3b − 4)
---
**Question 5:**
To factor \(81x^4 − 16\):
1. Recognize it as a difference of squares: \( (9x^2)^2 - 4^2\)
2. Result: \((9x^2 - 4)(9x^2 + 4)\)
3. Factor \((9x^2 - 4)\) again: \( (3x - 2)(3x + 2)(9x^2 + 4)\)
**Answer:** (3x − 2)(3x + 2)(9x^2 + 4)
---
**Question 6:**
To confirm a perfect square trinomial:
- \(16x^2 − 56x + 49 = (4x - 7)^2\)
- Check the others: They do not satisfy the conditions for perfect square trinomial.
**Answer:** 16x^2 − 56x + 49
---
**Question 7:**
To factor \(28a^3b^4 + 20a^2b^2 − 16ab^3\):
1. GCF is \(4ab\).
2. Result after factoring out GCF: \(4ab(7a^2b^3 + 5a - 4b^2)\)
**Answer:** 4ab(7a^2b + 5a − 4b^2)
---
**Question 8:**
To factor \(-4y^2 + 12y - 16\):
1. Factor out \(-4\): \(-4(y^2 - 3y + 4)\)
**Answer:** −4(y^2 − 3y + 4)
---
**Question 9:**
To factor \(2x^3 + 14x^2 + 6x + 42\):
1. Group: \(2x(x^2 + 7x + 3) + 14\)
2. Factor out GCF: \(2[(x^2 + 7)(x + 3)]\)
**Answer:** 2[(x2 + 3)(x + 7)]
---
**Question 10:**
\(x^2 + 25y^2\) is not factorable as it is the sum of squares.
**Answer:** Prime
---
**Question 11:**
To factor \(2x^2 − 6x − 36\):
1. Factor out \(2\): \(2(x^2 - 3x - 18)\)
2. Factor by grouping: \(x^2 - 3x - 18 = (x - 6)(x + 3)\)
**Answer:** 2(x − 3)(x + 6)
---
**Question 12:**
To find the side length of the square garden:
- Given \(4x^2 + 28x + 49 = (2x + 7)^2\)
**Answer:** (2x + 7) feet
---
**Question 13:**
To factor \(x^2 − 7x − 10\):
1. Find two numbers that multiply to \(-10\) and add to \(-7\): \( -10, +1 \)
2. Resulting factors: \((x - 10)(x + 1)\)
**Answer:** (x + 2)(x − 5)
---
**Question 14:**
To factor \(4x^2 + 23x − 6\):
1. Product-sum method gives pairs multiplying to \(-24\) and adding to \(23\).
2. Result: \((4x − 1)(x + 6)\)
**Answer:** (4x − 1)(x + 6)
---
**Question 15:**
To factor \(3x^2 − 30x + 63\):
1. Factor out \(3\): \(3(x^2 - 10x + 21)\)
2. Find pairs: \((x - 7)(x - 3)\)
**Answer:** 3(x − 7)(x − 3)
---
If you have any additional questions or need further clarification on any topic, feel free to ask!
**Question 1:**
The remaining factor that completes the equation \(x^2 - 25 = (x + 5)(\_\_\_\_) \) is the difference of squares:
1. \(x^2 - 25 = (x + 5)(x - 5)\)
**Answer:** (x − 5)
---
**Question 2:**
To factor \(6x^2 + 2xy + 9x + 3y\), we can group terms:
- Factor by grouping: \(2x(3x + y) + 3(3x + y) \)
- Result: \((2x + 3)(3x + y)\)
**Answer:** (2x + 3)(3x + y)
---
**Question 3:**
To factor \(x^3 + 6x^2 - 9x - 54\):
1. Grouping: \((x^3 + 6x^2) + (-9x - 54)\)
2. Factor out: \(x^2(x + 6) - 9(x + 6)\)
3. Result: \((x + 6)(x^2 - 9)\)
4. \(x^2 - 9\) is a difference of squares so: \((x + 6)(x + 3)(x - 3)\)
**Answer:** (x + 3)(x − 3)(x + 6)
---
**Question 4:**
To factor \(6ab - 8a + 21b - 28\):
1. Grouping: \((6ab - 8a) + (21b - 28)\)
2. Factor out: \(2a(3b - 4) + 7(3b - 4)\)
3. Result: \((2a + 7)(3b - 4)\)
**Answer:** (2a + 7)(3b − 4)
---
**Question 5:**
To factor \(81x^4 − 16\):
1. Recognize it as a difference of squares: \( (9x^2)^2 - 4^2\)
2. Result: \((9x^2 - 4)(9x^2 + 4)\)
3. Factor \((9x^2 - 4)\) again: \( (3x - 2)(3x + 2)(9x^2 + 4)\)
**Answer:** (3x − 2)(3x + 2)(9x^2 + 4)
---
**Question 6:**
To confirm a perfect square trinomial:
- \(16x^2 − 56x + 49 = (4x - 7)^2\)
- Check the others: They do not satisfy the conditions for perfect square trinomial.
**Answer:** 16x^2 − 56x + 49
---
**Question 7:**
To factor \(28a^3b^4 + 20a^2b^2 − 16ab^3\):
1. GCF is \(4ab\).
2. Result after factoring out GCF: \(4ab(7a^2b^3 + 5a - 4b^2)\)
**Answer:** 4ab(7a^2b + 5a − 4b^2)
---
**Question 8:**
To factor \(-4y^2 + 12y - 16\):
1. Factor out \(-4\): \(-4(y^2 - 3y + 4)\)
**Answer:** −4(y^2 − 3y + 4)
---
**Question 9:**
To factor \(2x^3 + 14x^2 + 6x + 42\):
1. Group: \(2x(x^2 + 7x + 3) + 14\)
2. Factor out GCF: \(2[(x^2 + 7)(x + 3)]\)
**Answer:** 2[(x2 + 3)(x + 7)]
---
**Question 10:**
\(x^2 + 25y^2\) is not factorable as it is the sum of squares.
**Answer:** Prime
---
**Question 11:**
To factor \(2x^2 − 6x − 36\):
1. Factor out \(2\): \(2(x^2 - 3x - 18)\)
2. Factor by grouping: \(x^2 - 3x - 18 = (x - 6)(x + 3)\)
**Answer:** 2(x − 3)(x + 6)
---
**Question 12:**
To find the side length of the square garden:
- Given \(4x^2 + 28x + 49 = (2x + 7)^2\)
**Answer:** (2x + 7) feet
---
**Question 13:**
To factor \(x^2 − 7x − 10\):
1. Find two numbers that multiply to \(-10\) and add to \(-7\): \( -10, +1 \)
2. Resulting factors: \((x - 10)(x + 1)\)
**Answer:** (x + 2)(x − 5)
---
**Question 14:**
To factor \(4x^2 + 23x − 6\):
1. Product-sum method gives pairs multiplying to \(-24\) and adding to \(23\).
2. Result: \((4x − 1)(x + 6)\)
**Answer:** (4x − 1)(x + 6)
---
**Question 15:**
To factor \(3x^2 − 30x + 63\):
1. Factor out \(3\): \(3(x^2 - 10x + 21)\)
2. Find pairs: \((x - 7)(x - 3)\)
**Answer:** 3(x − 7)(x − 3)
---
If you have any additional questions or need further clarification on any topic, feel free to ask!