In Mathematica, how can I cut off the high-order terms in a polynomial?

Question

For example, I have a polynomial y＝a_0＋a_1 x + ..... + a_50 x^50. Since I know that the high-order terms are imposing negligible effects on the evaluation of y, I want to cut off them and have something like y＝a_0＋a_1 x + ..... + a_10 x^10, the first eleven terms. How can I realize this?

I thank you all in advance.

Answer 1

In[1]:= y = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4;
y /. x^b_ /; b >= 3 -> 0

Out[2]= a0 + a1 x + a2 x^2

Answer 2

The mathematically proper approach..

  Series[ a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4, {x, 0, 2}] // Normal

  -> a0 + a1 x + a2 x^2

Answer 3

If your polynomial is actually as simple as shown, with a term for every power of x and none others, you can simply use Take or Part to extract only those terms that you want because of the automatic ordering (in Plus) that Mathematica uses. For example:

exp1 = Expand[(1 + x)^9]

Take[exp1, 5]

1 + 9 x + 36 x^2 + 84 x^3 + 126 x^4 + 126 x^5 + 84 x^6 + 36 x^7 + 9 x^8 + x^9

1 + 9 x + 36 x^2 + 84 x^3 + 126 x^4

If it is not you will need something else. Bill's replacement rule is one concise and efficient method. For more complex manipulations you may wish to decompose the polynomial using CoefficientArrays, CoefficientRules, or CoefficientList.

Answer 4

There is a shortcut to the previous answers which is even more symbolic. You write, say,

y[x_] = a0 + a1 x + a2 x^2 + a3 x^3 + a4 x^4 + a5 x^5;
y[x] + O[x]^3

which gives you,

a0 + a1 x + a2 x^2 + O[x]^3