Answer:
Problem 1
- Solution:
1. Group the terms: (nm - np) + (2m - 2p)
2. Factor out the greatest common factor from each group: n(m - p) + 2(m - p)
3. Notice that the two binomials are the same. Factor out m - p: (m - p)(n + 2)
- Final Answer: (m - p)(n + 2)
Problem 2
- Solution:
1. Group the terms: (ad - bc) + (-bd + ac)
2. Factor out the greatest common factor from each group: d(a - b) + c(-b + a)
3. Notice that the two binomials are the same but in reverse order. Factor out a - b: (a - b)(d + c)
- Final Answer: (a - b)(d + c)
Problem 3
- Solution:
1. Group the terms: (z³ + z²) + (z + 1)
2. Factor out the greatest common factor from each group: z²(z + 1) + 1(z + 1)
3. Notice that the two binomials are the same. Factor out z + 1: (z + 1)(z² + 1)
- Final Answer: (z + 1)(z² + 1)
Problem 4
- Solution:
1. Group the terms: (3v + 15) + (uv + 3u)
2. Factor out the greatest common factor from each group: 3(v + 5) + u(v + 3)
3. Notice that the two binomials are not the same. Therefore, this polynomial cannot be factored by grouping.
- Final Answer: Cannot be factored by grouping.
Problem 5
- Solution:
1. Group the terms: (2xz + 6y) + (2xy + 6z)
2. Factor out the greatest common factor from each group: 2(xz + 3y) + 2(xy + 3z)
3. Notice that the two binomials are not the same. Therefore, this polynomial cannot be factored by grouping.
- Final Answer: Cannot be factored by grouping.
