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What is Quadratic Form of a Polynomial Equation?

What is Quadratic Form?

Summary

  1. x4-13x3+36 is a 4th degree polynomial
  2. 'u' is an expression in 'x' that we can substitute into a higher degree polynomial to make it look like a quadratic
  3. u2-13u+36 is x4-13x2+36 rewritten in quadratic form
  4. You can solve equations in quadratic form just like regular quadratic equations
  5. u2=(x2)2 which equals x4
  6. 'a', 'b', and 'c' must all be real numbers, and 'a' cannot equal 0
  7. 'u' doesn't always have to equal 'x2'
  8. For example, to put '4x6-6x3+5' in quadratic form, you would let 'u=2x3'

Notes

    1. x4-13x3+36 is a 4th degree polynomial
    2. We know how to solve quadratic equations, so let's find a way to rewrite this to look like a quadratic
    1. Quadratics are 2nd degree polynomials, which means their highest power term is an x2 term
    2. We can rewrite the term x4 to look sort of like a 2nd degree term:
    3. x4 = x(2•2) = (x2)2
    4. Substituting 'u' for 'x2' makes the 4th and 2nd degree terms look like 2nd and 1st degree terms
    5. It looks like a quadratic equation with three terms that's easy to solve!
    1. A quadratic is a 2nd degree polynomial
    2. au2+bu+c is called the quadratic form of the equivalent expression in 'x'
    3. 'u' is an expression in terms of 'x' that we can substitute for 'x' to make the polynomial look like a quadratic
    1. A quadratic is a 2nd degree polynomial
    2. 'a', 'b', and 'c' are constant coefficients of each term
    3. If a=0, then we wouldn't have a quadratic because 0•u2=0, and we need a squared term to have a quadratic
    4. An 'expression in x' means that a substitute variable, like 'u', represents an algebraic expression with variable 'x' in it
    1. A quadratic is a 2nd degree polynomial
    2. To be quadratic in form, you MUST be able to find an expression in x that will make the polynomial appear to be a quadratic
    3. An 'expression in x' means that a substitute variable, like 'u', represents an algebraic expression with variable 'x' in it
    4. x3-x2+6 is an example of a polynomial that CANNOT be written in quadratic form
    1. x4+5x2+3 is a 4th degree polynomial
    2. Substituting 'u' for 'x2' puts this polynomial in quadratic form
    3. 'u' just needs to be an expression with 'x' in it that will make the polynomial look like a quadratic when we substitute it in
    4. For example, to get the quadratic form of '4x6-6x3+5', let 'u=2x3'
    1. Remember, when you're solving a quadratic equation you're solving for 'x'
    2. So after you find the answer in 'u', you'll have to substitute the 'x' expression back in for 'u'
    3. Then you can finish solving for 'x'
    1. This may help you remember what to do
    2. To go to a Halloween party, you need to change into a costume
    3. At the party you appear to be a ghost
    4. 'u' disguises an expression in 'x' just like a costume masks who you are
    5. But you have to remove the disguise to get back to reality