FACTORING POLYNOMIALS USING SYNTHETIC DIVISION

In this section, we use the synthetic division method that helps to factor a cubic polynomial into linear factors. If we identify one linear factor of cubic polynomial p(x) then using synthetic division we can get the quadratic factor of p(x). Further if possible one can factor the quadratic factor into linear factors.

Factor each of the following polynomials using synthetic division :

Example 1 :

x3 - 3x2 - 10x + 24

Solution :

By Substituting x = 2, we get the remainder 0.

So (x - 2) is a factor.

Then, 

x2 - x - 12 = x2 - 4x + 3x - 12

x2 - x - 12 = x(x - 4) + 3(x - 4)

x2 - x - 12 = (x + 3)(x - 4)

Therefore, the factors are (x - 2)(x + 3)(x- 4).

Example 2 :

2x3 - 3x2 - 3x + 2

Solution :

By substituting x = -1, we get the remainder 0.

So (x + 1) is a factor.

Then,

2x2 - 5x + 2 = 2x2 - 4x - x + 2

2x2 - 5x + 2 = 2x(x - 2) - 1(x - 2)

2x2 - 5x + 2 = (2x - 1)(x - 2)

Therefore, the factors are (x + 1)(2x - 1)(x - 2).

Example 3 :

-7x + 3 + 4x3

Solution :

-7x + 3 + 4x3  =  4x3 + 0x2 - 7x + 3

By substituting x = 1, we get the remainder 0.

So (x - 1) is a factor.

Then,

4x2 + 4x - 3 = 4x2 + 6x - 2x - 3

4x2 + 4x - 3 = 2x(2x + 3) - 1(2x + 3)

4x2 + 4x - 3 = (2x - 1)(2x + 3)

Therefore, the factors are (x - 1)(2x - 1)(2x + 3).

Example 4 :

x3 + x2 - 14x - 24

Solution :

By substituting x = -2, we get the remainder 0.

So (x + 2) is a factor.

Then,

x2 - x - 12 = x2 - 4x  + 3x - 12

x2 - x - 12 = x(x - 4) + 3(x - 4)

x2 - x - 12 = (x + 3)(x - 4)

Therefore, the factors are (x + 2)(x + 3)(x - 4).

Example 5 :

x3  - 7x + 6

Solution :

By substituting x = 1, we get the remainder 0.

So (x - 1) is a factor.

Then,

x2 + x - 6 = x2 + 3x  - 2x - 6

x2 + x - 6 = x(x + 3) - 2(x + 3)

x2 + x - 6 = (x + 3)(x - 2)

Therefore, the factors are (x - 2)(x + 3)(x - 1).

Example 6 :

x3 - 10x2 - x + 10

Solution :

By substituting x = 1, we get the remainder 0.

So (x - 1) is a factor.

Then,

x2 - 9x - 10 = x2 - 10x + 1x - 10

x2 - 9x - 10 = x(x - 10) + 1(x - 10)

x2 - 9x - 10 = (x + 1)(x - 10)

Therefore, the factors are (x + 1)(x - 10)(x - 1).

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