Determine intersections without polynomial division?

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In school, the question of the intersection of two graphs is often asked. There is a simpler method to calculate intersections without polynomial division, but this is usually not even discussed in class.

Understand the numbers
Understand the numbers

Calculation of the intersection of two straight lines without polynomial division

  • Calculating the intersection of two Straight lines only works without polynomial division, since this is not necessary here. Instead, just set the two straight lines equal and solve for the unknown variable.
  • The result is usually only the solution for the x value of the intersection. To determine the y coordinate of the intersection, insert the calculated x value into one of the two straight line equations and solve for y.

Calculation of the intersections of polynomials of degree n

  1. Since the polynomial division is a very complex process, the Horner scheme can also simply be used to calculate intersection points. First, draw a table with three columns on the paper. Then you set your two polynomials equal and bring everything to one side so that the other side of the equation is zero. Grasp the same Potencies together and arrange the equation in order of decreasing power.
  2. Guess a root of the equation. In school this is often a number that is divisible by the last coefficient without a trailing x. Write this zero in the bottom column of the table in the left margin and separate the column with a vertical line.
  3. In the top line to the right of the line, write down all the x-powers one after the other. Below you write the corresponding coefficient. If there is no number in front of an x, the coefficient is 1. Please also note the sign. Separate the x-powers with their coefficients by vertical lines and write the number zero next to the last coefficient.
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  4. The Horner scheme says you need to write the first coefficient one column to the right below the second coefficient. The cell under the first coefficient remains empty. Now multiply the number you just wrote down by the guessed zero and add the result to the coefficient above. Note the result in the bottom line next to the last result and carry out this step until you have reached the end of the table.
  5. Write down the coefficients of the last row one after the other with the corresponding x-powers. You get the same result as with a complex polynomial division. If you come to a result whose degree is higher than 2, you can run the Horner scheme again on the result. Otherwise, you can get the solutions by using the solution formula. This is: x1 = (-b + ROOT (b ^ 2-4ac)) / (2 * a); x2 = (-b-ROOT (b ^ 2-4ac)) / (2 * a). The letters in alphabetical order correspond to the coefficients of a polynomial of the second degree.
  6. But with this you have only calculated the x values ​​of the intersection. Insert the results into one of the polynomials in turn. This gives you the associated y value and thus the complete intersection points.

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