Question

factor each completely.

1. $12p^{3}+28p^{2}+24p+56$
2. $112x^{3}-64x^{2}+140x-80$
3. $25v^{3}-35v^{2}+35v-49$
4. $14v^{3}+35v^{2}+4v+10$
5. $9m^{4}-87m^{2}-30$
6. $2a^{4}-17a^{2}+21$

Explanation:

Problem 17: $12p^3 + 28p^2 + 24p + 56$

Step1: Group terms into pairs

$(12p^3 + 28p^2) + (24p + 56)$

Step2: Factor GCF from each pair

$4p^2(3p + 7) + 8(3p + 7)$

Step3: Factor out common binomial

$(3p + 7)(4p^2 + 8)$

Step4: Factor GCF from second term

$(3p + 7) \cdot 4(p^2 + 2) = 4(3p + 7)(p^2 + 2)$

---

Problem 18: $112x^3 - 64x^2 + 140x - 80$

Step1: Group terms into pairs

$(112x^3 - 64x^2) + (140x - 80)$

Step2: Factor GCF from each pair

$16x^2(7x - 4) + 20(7x - 4)$

Step3: Factor out common binomial

$(7x - 4)(16x^2 + 20)$

Step4: Factor GCF from second term

$(7x - 4) \cdot 4(4x^2 + 5) = 4(7x - 4)(4x^2 + 5)$

---

Problem 19: $25v^3 - 35v^2 + 35v - 49$

Step1: Group terms into pairs

$(25v^3 - 35v^2) + (35v - 49)$

Step2: Factor GCF from each pair

$5v^2(5v - 7) + 7(5v - 7)$

Step3: Factor out common binomial

$(5v - 7)(5v^2 + 7)$

---

Problem 20: $14v^3 + 35v^2 + 4v + 10$

Step1: Group terms into pairs

$(14v^3 + 35v^2) + (4v + 10)$

Step2: Factor GCF from each pair

$7v^2(2v + 5) + 2(2v + 5)$

Step3: Factor out common binomial

$(2v + 5)(7v^2 + 2)$

---

Problem 21: $9m^4 - 87m^2 - 30$

Step1: Factor out overall GCF

$3(3m^4 - 29m^2 - 10)$

Step2: Substitute $u=m^2$, factor quadratic

$3(3u^2 - 29u - 10) = 3(3u + 1)(u - 10)$

Step3: Substitute back $u=m^2$

$3(3m^2 + 1)(m^2 - 10)$

---

Problem 22: $2a^4 - 17a^2 + 21$

Step1: Substitute $u=a^2$, factor quadratic

$2u^2 - 17u + 21 = (2u - 3)(u - 7)$

Step2: Substitute back $u=a^2$

$(2a^2 - 3)(a^2 - 7)$

Answer:

1. $4(3p + 7)(p^2 + 2)$
2. $4(7x - 4)(4x^2 + 5)$
3. $(5v - 7)(5v^2 + 7)$
4. $(2v + 5)(7v^2 + 2)$
5. $3(3m^2 + 1)(m^2 - 10)$
6. $(2a^2 - 3)(a^2 - 7)$