To factor a polynomial of degree 3 or more, we can use synthetic division method.
In this method, we will find the factors of a polynomial by trial and error.
To learn synthetic division step by step, click here.
Example 1 :
Factor the following polynomial given that the product of two of the zeros is 8.
x4 + 2x3 - 25x2 - 26x + 120
Solution :
Because the product two of the zeros is 8, we can try 2 and 4 in synthetic division.
x = 2 and x = 4 are the two zeros of the given polynomial of degree 4.
Because x = 2 and x = 4 are the two zeros of the given polynomial, the two factors are (x - 2) and (x - 4).
To find other factors, factor the quadratic expression which has the coefficients 1, 8 and 15.
That is, x2 + 8x + 15.
x2 + 8x + 15 = (x + 3)(x + 5)
So, the factors of the given polynomial are
(x - 2), (x - 4), (x + 3) and (x + 5)
Example 2 :
Factor :
x4 - 10x3 + 37x2 - 60x + 36
Solution :
By trial and error, we can check whether 1 is a zero of the above polynomial.
Because the remainder is 4 (not zero), 1 is not a zero of the given polynomial.
Now, let us check with -1.
Because the remainder is 24 (not zero), -1 is not a zero of the given polynomial.
Now, let us check with 2.
Both x = 2 and x = 3 are the two zeros of the given polynomial.
Because x = 2 and x = 3 are the two zeros of the given polynomial, the two factors are (x - 2) and (x - 3).
To find other factors, factor the quadratic expression which has the coefficients 1, -5 and 6.
That is, x2 - 5x + 6.
x2 - 5x + 6 = (x - 2)(x - 3)
So, the factors of the given polynomial are
(x - 2), (x - 3), (x - 2) and (x - 3)
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