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Math 409/409G History of Mathematics

Math 409/409G History of Mathematics. Depressed Polynomial Equations. What’s a depressed polynomial equation?. An n th degree polynomial equation is said to be depressed if it is missing the ( n – 1) st term. For example: x 2 – 9  0 x 3 + 8 x  9 x 4 – 10 x 2 + 4 x + 8  0.

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Math 409/409G History of Mathematics

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  1. Math 409/409GHistory of Mathematics Depressed Polynomial Equations

  2. What’s a depressed polynomial equation? An nth degree polynomial equation is said to be depressed if it is missing the (n – 1)st term. For example: x2 – 9  0 x3 + 8x  9 x4 – 10x2 + 4x + 8  0

  3. A depressed quadratic equation is quite simple to solve. And as you will see in later, there are techniques for solving depressed cubic and quartic equations.

  4. Depressing an Equation Substituting x y – (b/na) in the equation will result in a nth degree, depressed equation in the variable y. Once the depressed equation is solved, the substitution x y – (b/na) can then be used to solve for x.

  5. Here’s what the substitution x y – (b/2a) does to a quadratic equation.

  6. Since we substituted x y – b/2a, the solution to the quadratic equation ax2 + bx + c  0 is

  7. Solve x3 + 6x2 + 3x 10 Making the substitution x y – 6/3·1,

  8. Solve the quarticx4 +12x3 + 49x2 + 70x + 40  0 Making the substitution x y – 12/4·1,

  9. Not all cubic and quartic equations can be solved by solving the depressed equation as we did in the last two examples. It’s usually the case that the depressed equation can’t be solved using the techniques you learned in high school. In the next lesson you will see how to solve any depressed cubic equation.

  10. This ends the lesson on Depressed Polynomial Equations

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