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Quartic Trinomial – Definition & Examples

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A quartic trinomial is defined as a polynomial with three terms having the highest degree 4. Some examples of quadratic trinomials are:

  1. x^4 + 17x + 16
  2. y^4 - 3y^2 + 1. In this example, the first and third degree terms are absent
  3. m^4 + 3m^2 + 8m. There is no constant term in this quartic trinomial.
  4. 8p^4 - p - 13

Steps for Factoring/Solving a Quartic Trinomial:

  1. Arrange the terms of the trinomial from highest to lowest.
  2. Check that the trinomial consists of the x^4 term, the x^2 term (middle term) and the constant term.
  3. Write down the middle term as a sum of two quantities, such that their product is equal to the product of the first term and the last term. This can be made clear by looking at the following two examples.

Example 1:

Consider the fourth degree trinomial given as , x^4 + 5x^2 + 4 As explained in step 3 above we write the middle term 5x^2 as 5x^2 = 4x^2 + x^2. We then have that, \begin{align*}x^4 + 5x^2 + 4 &= x^4 + 4x^2 + x^2 + 4 \\ &= x^2(x^2 + 4) + 1(x^2 + 4) \\ &= (x^2+1)(x^2 + 4)\end{align*} Hence we have obtained the required factorization.

Example 2:

Consider the fourth degree trinomial given as , x^4 - 2x^2 - 8 Note the the product of -4 and +2 is equal to -8 and the sum of the two numbers is equal to -2. Therefore we write the middle term - 2x^2 as - 2x^2 = -4x^2 + 2x^2. We then have that, \begin{align*}x^4 - 2x^2 - 8 &= x^4 - 4x^2 + 2x^2 - 8 \\ &= x^2(x^2 - 4) + 2(x^2 - 4) \\ &= (x^2+2)(x^2 - 4)\end{align*} Thus we have written the given quartic trinomial as a product of two factors. Infact we can further factor x^2 - 4 = x^2 - 2^2 using the formula a^2 - b^2 = (a+b)(a-b) . So we have that, \begin{align*}x^4 - 2x^2 - 8 &= (x^2+2)(x^2 - 4)\\ &= (x^2+2)(x^2 - 2^2) \\ &= (x^2+2)(x+2)(x-2) \end{align*}

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