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Study Guides > College Algebra

Section Exercises

1. Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false. 2. Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial. 3. You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps? 4. State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial. For the following exercises, identify the degree of the polynomial. 5. [latex]7x - 2{x}^{2}+13[/latex] 6. [latex]14{m}^{3}+{m}^{2}-16m+8[/latex] 7. [latex]-625{a}^{8}+16{b}^{4}[/latex] 8. [latex]200p - 30{p}^{2}m+40{m}^{3}[/latex] 9. [latex]{x}^{2}+4x+4[/latex] 10. [latex]6{y}^{4}-{y}^{5}+3y - 4[/latex] For the following exercises, find the sum or difference. 11. [latex]\left(12{x}^{2}+3x\right)-\left(8{x}^{2}-19\right)[/latex] 12. [latex]\left(4{z}^{3}+8{z}^{2}-z\right)+\left(-2{z}^{2}+z+6\right)[/latex] 13. [latex]\left(6{w}^{2}+24w+24\right)-\left(3w{}^{2}-6w+3\right)[/latex] 14. [latex]\left(7{a}^{3}+6{a}^{2}-4a - 13\right)+\left(-3{a}^{3}-4{a}^{2}+6a+17\right)[/latex] 15. [latex]\left(11{b}^{4}-6{b}^{3}+18{b}^{2}-4b+8\right)-\left(3{b}^{3}+6{b}^{2}+3b\right)[/latex] 16. [latex]\left(49{p}^{2}-25\right)+\left(16{p}^{4}-32{p}^{2}+16\right)[/latex] For the following exercises, find the product. 17. [latex]\left(4x+2\right)\left(6x - 4\right)[/latex] 18. [latex]\left(14{c}^{2}+4c\right)\left(2{c}^{2}-3c\right)[/latex] 19. [latex]\left(6{b}^{2}-6\right)\left(4{b}^{2}-4\right)[/latex] 20. [latex]\left(3d - 5\right)\left(2d+9\right)[/latex] 21. [latex]\left(9v - 11\right)\left(11v - 9\right)[/latex] 22. [latex]\left(4{t}^{2}+7t\right)\left(-3{t}^{2}+4\right)[/latex] 23. [latex]\left(8n - 4\right)\left({n}^{2}+9\right)[/latex] For the following exercises, expand the binomial. 24. [latex]{\left(4x+5\right)}^{2}[/latex] 25. [latex]{\left(3y - 7\right)}^{2}[/latex] 26. [latex]{\left(12 - 4x\right)}^{2}[/latex] 27. [latex]{\left(4p+9\right)}^{2}[/latex] 28. [latex]{\left(2m - 3\right)}^{2}[/latex] 29. [latex]{\left(3y - 6\right)}^{2}[/latex] 30. [latex]{\left(9b+1\right)}^{2}[/latex] For the following exercises, multiply the binomials. 31. [latex]\left(4c+1\right)\left(4c - 1\right)[/latex] 32. [latex]\left(9a - 4\right)\left(9a+4\right)[/latex] 33. [latex]\left(15n - 6\right)\left(15n+6\right)[/latex] 34. [latex]\left(25b+2\right)\left(25b - 2\right)[/latex] 35. [latex]\left(4+4m\right)\left(4 - 4m\right)[/latex] 36. [latex]\left(14p+7\right)\left(14p - 7\right)[/latex] 37. [latex]\left(11q - 10\right)\left(11q+10\right)[/latex] For the following exercises, multiply the polynomials. 38. [latex]\left(2{x}^{2}+2x+1\right)\left(4x - 1\right)[/latex] 39. [latex]\left(4{t}^{2}+t - 7\right)\left(4{t}^{2}-1\right)[/latex] 40. [latex]\left(x - 1\right)\left({x}^{2}-2x+1\right)[/latex] 41. [latex]\left(y - 2\right)\left({y}^{2}-4y - 9\right)[/latex] 42. [latex]\left(6k - 5\right)\left(6{k}^{2}+5k - 1\right)[/latex] 43. [latex]\left(3{p}^{2}+2p - 10\right)\left(p - 1\right)[/latex] 44. [latex]\left(4m - 13\right)\left(2{m}^{2}-7m+9\right)[/latex] 45. [latex]\left(a+b\right)\left(a-b\right)[/latex] 46. [latex]\left(4x - 6y\right)\left(6x - 4y\right)[/latex] 47. [latex]{\left(4t - 5u\right)}^{2}[/latex] 48. [latex]\left(9m+4n - 1\right)\left(2m+8\right)[/latex] 49. [latex]\left(4t-x\right)\left(t-x+1\right)[/latex] 50. [latex]\left({b}^{2}-1\right)\left({a}^{2}+2ab+{b}^{2}\right)[/latex] 51. [latex]\left(4r-d\right)\left(6r+7d\right)[/latex] 52. [latex]\left(x+y\right)\left({x}^{2}-xy+{y}^{2}\right)[/latex] 53. A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: [latex]\left(4x+1\right)\left(8x - 3\right)[/latex] where x is measured in meters. Multiply the binomials to find the area of the plot in standard form. 54. A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is [latex]{\left(2x+9\right)}^{2}[/latex]. The height of the silo is [latex]10x+10[/latex], where x is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold. For the following exercises, perform the given operations. 55. [latex]{\left(4t - 7\right)}^{2}\left(2t+1\right)-\left(4{t}^{2}+2t+11\right)[/latex] 56. [latex]\left(3b+6\right)\left(3b - 6\right)\left(9{b}^{2}-36\right)[/latex] 57. [latex]\left({a}^{2}+4ac+4{c}^{2}\right)\left({a}^{2}-4{c}^{2}\right)[/latex]

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