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. \(x - 7\sqrt x - 18\)