Factoring By Grouping Date $$ \qquad $$ Period Factor each completely. 1) $$ 12 a^{3}-9 a^{2}+4 a-3 $$ 2) $$ 2 p^{3}+5 p^{2}+6 p+15 $$ 3) $$ 3 n^{3}-4 n^{2}+9 n-12 $$ 4) $$ 12 n^{3}+4 n^{2}+3 n+1 $$ 5) $$ m^{3}-m^{2}+2 m-2 $$ 6) $$ 5 n^{3}-10 n^{2}+3 n-6 $$ 7) $$ 35 x y-5 x-56 y+8 $$ 8) $$ 224 a z+56 a c-84 y z-21 y c $$

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UpStudy AI · Accepted Answer

Sure, let's factor each polynomial by grouping step by step.

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### **1) $$ 12a^{3} - 9a^{2} + 4a - 3 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(12a^3 - 9a^2) + (4a - 3)$$
2. **Factor out the greatest common factor (GCF) from each group:**  
   $$3a^2(4a - 3) + 1(4a - 3)$$
3. **Factor out the common binomial:**  
   $$(3a^2 + 1)(4a - 3)$$

**Factored Form:**  
$$
\boxed{(3a^{2} + 1)(4a - 3)}
$$

---

### **2) $$ 2p^{3} + 5p^{2} + 6p + 15 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(2p^3 + 5p^2) + (6p + 15)$$
2. **Factor out the GCF from each group:**  
   $$p^2(2p + 5) + 3(2p + 5)$$
3. **Factor out the common binomial:**  
   $$(p^2 + 3)(2p + 5)$$

**Factored Form:**  
$$
\boxed{(p^{2} + 3)(2p + 5)}
$$

---

### **3) $$ 3n^{3} - 4n^{2} + 9n - 12 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(3n^3 - 4n^2) + (9n - 12)$$
2. **Factor out the GCF from each group:**  
   $$n^2(3n - 4) + 3(3n - 4)$$
3. **Factor out the common binomial:**  
   $$(n^2 + 3)(3n - 4)$$

**Factored Form:**  
$$
\boxed{(n^{2} + 3)(3n - 4)}
$$

---

### **4) $$ 12n^{3} + 4n^{2} + 3n + 1 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(12n^3 + 4n^2) + (3n + 1)$$
2. **Factor out the GCF from each group:**  
   $$4n^2(3n + 1) + 1(3n + 1)$$
3. **Factor out the common binomial:**  
   $$(4n^2 + 1)(3n + 1)$$

**Factored Form:**  
$$
\boxed{(4n^{2} + 1)(3n + 1)}
$$

---

### **5) $$ m^{3} - m^{2} + 2m - 2 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(m^3 - m^2) + (2m - 2)$$
2. **Factor out the GCF from each group:**  
   $$m^2(m - 1) + 2(m - 1)$$
3. **Factor out the common binomial:**  
   $$(m^2 + 2)(m - 1)$$

**Factored Form:**  
$$
\boxed{(m^{2} + 2)(m - 1)}
$$

---

### **6) $$ 5n^{3} - 10n^{2} + 3n - 6 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(5n^3 - 10n^2) + (3n - 6)$$
2. **Factor out the GCF from each group:**  
   $$5n^2(n - 2) + 3(n - 2)$$
3. **Factor out the common binomial:**  
   $$(5n^2 + 3)(n - 2)$$

**Factored Form:**  
$$
\boxed{(n - 2)(5n^{2} + 3)}
$$

---

### **7) $$ 35xy - 5x - 56yz + 8 $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(35xy - 5x) + (-56yz + 8)$$
2. **Factor out the GCF from each group:**  
   $$5x(7y - 1) - 8(7y - 1)$$
3. **Factor out the common binomial:**  
   $$(5x - 8)(7y - 1)$$

**Factored Form:**  
$$
\boxed{(5x - 8)(7y - 1)}
$$

---

### **8) $$ 224az + 56ac - 84yz - 21yc $$**

**Step-by-Step:**
1. **Group terms:**  
   $$(224az + 56ac) + (-84yz - 21yc)$$
2. **Factor out the GCF from each group:**  
   $$56a(4z + c) - 21y(4z + c)$$
3. **Factor out the common binomial:**  
   $$(56a - 21y)(4z + c)$$
4. **Factor out the GCF from the first binomial:**  
   $$7(8a - 3y)(4z + c)$$

**Factored Form:**  
$$
\boxed{7(8a - 3y)(4z + c)}
$$

---

Each polynomial has been successfully factored by grouping, resulting in the expressions above.
1. $$ (3a^{2} + 1)(4a - 3) $$ 2. $$ (p^{2} + 3)(2p + 5) $$ 3. $$ (n^{2} + 3)(3n - 4) $$ 4. $$ (4n^{2} + 1)(3n + 1) $$ 5. $$ (m^{2} + 2)(m - 1) $$ 6. $$ (n - 2)(5n^{2} + 3) $$ 7. $$ (5x - 8)(7y - 1) $$ 8. $$ 7(8a - 3y)(4z + c) $$

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