FACTORING 4TH DEGREE POLYNOMIALS

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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