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            Paul's Online Notes
            Paul's Online Notes
            Home / Algebra / Preliminaries / Polynomials
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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 \(n\)?
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