Factoring Polynomials Problems

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Rules for the factorization of the polynomials  are given below.

1. The first term of both the factor is x.
2. The product of the second terms of the two factors is equal to the third term of the trinomial.
3. The algebraic sum of the second terms of the two factors is equal to the coefficient of x in the trinomial.

Example 1. Resolve into factors   x2 + 11x + 24.         
The second term of the factors must be such that their product is + 24 and their sum + 11, it is clear that they must be + 8 and + 3.

x2 + 11x + 24
= x2 + 8 x + 3 x + 24
= x (x + 8) + 3 (x + 8)

Or   x2 + 11x + 24

=(x + 8) (x + 3)
 
Example 2. Resolve into factors  x2 - 10x + 24.

The second term of the factors must be such that their product is + 24 and their sum -10, it is clear that they must be - 6 and - 4.

x2 - 10x + 24
=   x2 - 6 x - 4 x + 24
=   x (x - 6) - 4 (x - 6)

Or   x2 - 10x + 24 = (x - 6) (x - 4)
 
Example 3. Resolve into factors  x2 + 2x - 35.

In the given equation the third term is negative. The second terms of the factors must be such that their product is – 35 and their algebraically sum + 2. Hence they must have opposite signs, and the greater of them must be negative in order to give its sign to their sum
= x2 + 2x – 35 
=  x2 + 7x- 5 x – 35        or
= x (x + 7) – 5 (x + 7)

Hence x2 + 2x – 35 = (x + 7) (x – 5)

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