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### Section 1-4 : Polynomials

For problems 1 – 18 perform the indicated operation and identify the degree of the result.

1. Add $$10{x^5} + 2{x^3} - 1$$ to $$8{x^4} - {x^3} + 16{x^2}$$
2. Add $$7{t^2} - 13t + 4$$ to $$- 6{t^2} + 13t - 4$$
3. Subtract $$- 12{z^2} + 9z - 3$$ from $${z^3} + 2{z^2} - 15z + 7$$
4. Subtract $$100{x^4} - 19{x^2} - 7x$$ from $$150{x^3} + 8x - 14$$
5. Subtract $${w^4} + {w^3} + {w^2} + w + 1$$ from $${w^5}$$
6. $$6{y^2}\left( {3 - {y^2} + 2{y^3}} \right)$$
7. $${x^9}\left( {{x^2} + 7x - 4} \right)$$
8. $$\left( {7x - 5} \right)\left( {4 - 10x} \right)$$
9. $$\left( {4 + 9{t^2}} \right)\left( {{t^3} - 3t} \right)$$
10. $$\left( {1 + 8y} \right)\left( {{y^3} - 4{y^2} + 7} \right)$$
11. $$7\left( {x - 9} \right)\left( {2x + 3} \right)$$
12. $${z^2}\left( {1 - {z^2}} \right)\left( {1 + {z^2}} \right)$$
13. $$\left( {2 - x + 4{x^2}} \right)\left( {6x + 7} \right)$$
14. $$\left( {10{w^2} - 4w + 9} \right)\left( {{w^3} + 5{w^2} + 2} \right)$$
15. $$10{\left( {x + 3{x^2}} \right)^2}$$
16. $$\left( {1 - 5y} \right){\left( {4 + y} \right)^2}$$
17. Subtract $$\left( {3 - x} \right)\left( {3 + x} \right)$$ from $${x^2} - 7x + 10$$
18. Subtract $${\left( {4{x^2} - 1} \right)^2}$$ from $${\left( {x + 9{x^3}} \right)^2}$$
19. If we multiply a polynomial with degree n and a polynomial of degree m what is the degree of the result?
20. If we add 2 polynomials of degree n and m with $$n < m$$ what is the degree of the result?
21. If we subtract 2 polynomials of degree n and m with $$n < m$$ what is the degree of the result?
22. If we add two polynomials, both of degree $$n$$, is it possible for the result to not be degree $$n$$? If it is not possible can you give an example of two polynomials, both of degree $$n$$, whose sum is not degree $$n$$?
23. If we subtract two polynomials, both of degree $$n$$, is it possible for the result to not be degree $$n$$? If it is not possible can you give an example of two polynomials, both of degree $$n$$, whose difference is not degree $$n$$?