## Understanding Polynomials: Polynomial Behavior, Turning Points, Why They Behave the Way They Do

Here’s a graph and chart I made to help you understand how polynomials and their turning points work. I think polynomials should always be taught this way: graphing the individual terms, and showing that they are literally the combination of all of the y values of all the terms. And all junior high and up students should be taught how to graph such terms and polynomials in the ubiquitous Microsoft Excel, not just graphic calculators–Excel should be a part of any modern junior high/high school math class as a stepping stone to lots of work they’ll be doing in the future in Excel and other more math-dedicated programs and languages.

The first column in the table below is the x value, the second column is the value of the whole polynomial 4(x-1)^3 – 12(x-1)^2 +15, the third column is the value of the first term 4(x-1)^3, the fourth column is the value of the second term -12(x-1)^2, and the fifth column is the value of the third term, the constant 15.

You can see per x value how the positiveness or negativeness of the first and second terms, plus the constant 15, how the size and sign of the first two terms combined with the constant 15 add up to make the net of the whole polynomial positive or negative (or zero) at different points on the x axis. I put in x axis grid lines so you could count off grid marks too see how for the same x value, sometimes the first term has a higher positive than the second has a negative, and vice versa (of course offset to a degree by the constant 15).

Chapter 8 of Algebra II for Dummies has some great info on polynomials and their turning points also. The chapter also has great information on finding roots of polynomials, patterns to look for in factoring, and the Rational Root Theorem, which basically helps you find rational roots (rational numbers have terminating decimals or decimals that repeat in a pattern). You may need a graphing calculator/computer to estimate irrational roots, as they don’t factor cleanly/evenly).

 x 4(x-1)^3 – 12(x-1)^2 +15 4(x-1)^3 -12(x-1)^2 +15 -8 -3873 -2916 -972 15 -7.75 -3583.4375 -2679.69 -918.75 15 -7.5 -3308.5 -2456.5 -867 15 -7.25 -3047.8125 -2246.06 -816.75 15 -7 -2801 -2048 -768 15 -6.75 -2567.6875 -1861.94 -720.75 15 -6.5 -2347.5 -1687.5 -675 15 -6.25 -2140.0625 -1524.31 -630.75 15 -6 -1945 -1372 -588 15 -5.75 -1761.9375 -1230.19 -546.75 15 -5.5 -1590.5 -1098.5 -507 15 -5.25 -1430.3125 -976.563 -468.75 15 -5 -1281 -864 -432 15 -4.75 -1142.1875 -760.438 -396.75 15 -4.5 -1013.5 -665.5 -363 15 -4.25 -894.5625 -578.813 -330.75 15 -4 -785 -500 -300 15 -3.75 -684.4375 -428.688 -270.75 15 -3.5 -592.5 -364.5 -243 15 -3.25 -508.8125 -307.063 -216.75 15 -3 -433 -256 -192 15 -2.75 -364.6875 -210.938 -168.75 15 -2.5 -303.5 -171.5 -147 15 -2.25 -249.0625 -137.313 -126.75 15 -2 -201 -108 -108 15 -1.75 -158.9375 -83.1875 -90.75 15 -1.5 -122.5 -62.5 -75 15 -1.25 -91.3125 -45.5625 -60.75 15 -1 -65 -32 -48 15 -0.75 -43.1875 -21.4375 -36.75 15 -0.5 -25.5 -13.5 -27 15 -0.25 -11.5625 -7.8125 -18.75 15 0 -1 -4 -12 15 0.25 6.5625 -1.6875 -6.75 15 0.5 11.5 -0.5 -3 15 0.75 14.1875 -0.0625 -0.75 15 1 15 0 0 15 1.25 14.3125 0.0625 -0.75 15 1.5 12.5 0.5 -3 15 1.75 9.9375 1.6875 -6.75 15 2 7 4 -12 15 2.25 4.0625 7.8125 -18.75 15 2.5 1.5 13.5 -27 15 2.75 -0.3125 21.4375 -36.75 15 3 -1 32 -48 15 3.25 -0.1875 45.5625 -60.75 15 3.5 2.5 62.5 -75 15 3.75 7.4375 83.1875 -90.75 15 4 15 108 -108 15 4.25 25.5625 137.3125 -126.75 15 4.5 39.5 171.5 -147 15 4.75 57.1875 210.9375 -168.75 15 5 79 256 -192 15 5.25 105.3125 307.0625 -216.75 15 5.5 136.5 364.5 -243 15 5.75 172.9375 428.6875 -270.75 15 6 215 500 -300 15 6.25 263.0625 578.8125 -330.75 15 6.5 317.5 665.5 -363 15 6.75 378.6875 760.4375 -396.75 15 7 447 864 -432 15 7.25 522.8125 976.5625 -468.75 15 7.5 606.5 1098.5 -507 15 7.75 698.4375 1230.188 -546.75 15 8 799 1372 -588 15

Internal tag: math