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### Section 1.5 : Factoring Polynomials

For problems 1 – 8 factor out the greatest common factor from each polynomial.

1. $${x^3} - 6{x^8} + 10{x^{10}}$$
2. $$25{u^6} - 15{u^5} + 30{u^8}$$
3. $$2{y^6}z - {y^4}{z^{10}} + 3{y^2}{z^2}$$
4. $$7{a^{10}}{b^7} + 14{a^8}{b^9} - 35{a^6}{b^{12}}$$
5. $$3\left( {9 + 7x} \right) - \left( {2 - x} \right)\left( {9 + 7x} \right)$$
6. $${z^2}\left( {4z - {z^3}} \right) + 7\left( {{z^3} - 4z} \right)$$
7. $$8y{\left( {2y + 7} \right)^4} - 2{y^3}{\left( {2y + 7} \right)^9}$$
8. $${w^2}\left( {1 + {w^2}} \right){\left( {8w - 1} \right)^{10}} + 9w{\left( {1 + {w^2}} \right)^4}{\left( {8w - 1} \right)^7}$$

For problems 9 – 13 factor each of the following by grouping.

1. $$18x - 2{x^3} + 9 - {x^2}$$
2. $$6{w^4} + 3{w^3} - 14{w^2} - 7w$$
3. $${y^4} + {y^3} + 9{y^3} + 9{y^2}$$
4. $$21x - 56{x^4} - 12{x^3} + 32{x^6}$$
5. $$6{t^3} + 3{t^4} - 2{t^5} - {t^6}$$

For problems 14 – 32 factor each of the following.

1. $${x^2} - 10x + 9$$
2. $${t^2} + 11t + 24$$
3. $${z^2} - 9z - 10$$
4. $${x^2} - 3x - 28$$
5. $${x^2} + 10x - 24$$
6. $${w^2} - 8w + 16$$
7. $${z^2} + 6z + 9$$
8. $${x^2} - 144$$
9. $$36 - {x^2}$$
10. $$4{z^2} - 23z - 6$$
11. $$2{y^2} - 9y + 10$$
12. $$12{x^2} + 31x + 7$$
13. $$6{z^2} - 35z + 36$$
14. $$8{t^2} + 29t - 12$$
15. $$21 - w - 2{w^2}$$
16. $$36{v^2} - 49$$
17. $$100{x^2} + 20x + 1$$
18. $$25{z^2} - 40z + 16$$
19. $$9{y^2} - 121$$

For problems 33 – 38 factor each of the following.

1. $$4{x^3} - 20{x^2} - 144x$$
2. $${t^4} + 15{t^3} + 14{t^2}$$
3. $$6{u^8} - 3{u^6} - 3{u^4}$$
4. $${t^8} + 5{t^4} - 24$$
5. $$2{z^4} - 5{z^2} - 12$$
6. $$4{x^6} + {x^3} - 5$$

For problems 39 & 40 determine the possible values of $$a$$ for which the polynomial will factor.

1. $${x^2} + ax - 16$$
2. $${x^2} + ax + 20$$

For problems 41 – 44 use the knowledge of factoring that you’ve learned in this section to factor the following expressions.

1. $${x^2} + 1 - 6{x^{ - 2}}$$
2. $$\displaystyle {x^2} - 2 + \frac{1}{{{x^2}}}$$
3. $${x^4} - \frac{{49}}{{{x^2}}}$$
4. $$x - 7\sqrt x - 18$$