Factoring 4th degree polynomials :
To factor a polynomial of degree 3 and greater than 3, we can to use the method called synthetic division method.
In this method we have to use trial and error to find the factors. This is one of the shortcut to find factors.
We also have another direct method to factorize a polynomial of degree 4. Let us see some example problems by using above methods.
Example 1 :
Solve the equation x⁴ + 2x³ - 25 x² - 26 x + 120 = 0 given that the product of two roots is 8.
Since the product two roots is 8, we can try 2 and 4 in synthetic division.
x = 2 and x = 4 are the two roots of the given polynomial of degree 4. To find other roots we have to factorize the quadratic equation x² + 8x + 15.
x² + 8x + 15 = (x + 3) (x + 5)
To find roots, we have to set the linear factors equal to zero.
(x + 3) (x + 5) = 0
x + 3 = 0
Subtract 3 on both sides
x + 3 - 3 = 0 - 3
x = -3
x + 5 = 0
Subtract 5 on both sides
x + 5 - 5 = 0 - 5
x = -5
Hence the roots are -3, -5,2 and 4.
Example 2 :
Solve the equation x⁴ - 10x³ + 37 x² - 60 x + 36 = 0
Let us use some trial and error to find one of its factors.
Now check whether 1 is a factor of the given polynomial
1 is not a root of the given polynomial. Now let us check with -1.
-1 is not a root of the given polynomial. Now let us check with 2.
2 and 3 are factors of the given polynomial. To find other two factors, we have to factorize the quadratic equation x² -5x + 6.
x² -5x + 6 = (x-2) (x-3)
x - 2 = 0
Add 2 on both sides
x - 2 + 2 = 0 + 2
x = 2
x - 3 = 0
Add 3 on both sides
x - 3 + 3 = 0 + 3
x = 3
Hence the roots are 2, 2,3 and 3.
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