Rational root test in Polynomial

Rational root test in Polynomial

a) Find the roots of constant term name them as P.

b) Find the roots of leading coefficient ( term with highest degree) name them as ‘q’.

c) Possible roots are ~+mn~ P/q

Examples :

1) Find the factors of 3x
3 - 4x 2 -17x +6

Solution :
Factors of 6 =P = 1,2,3,6

Factors of 3 = q =1 , 3

Possible factors are p/q = ~+mn~ 1,2,3,6,1/3,2/3

Now, use synthetic division with 1

As remainder is 0.

(x- 3) is one of the factor.

As in synthetic division numbers below are 3, 5 and -2 (three numbers)

So next polynomial will be 3x
2 + 5x -2

Find the factors of this using method of splitting term.

 Polynomial 3x2 + 5x -2 Multiply coefficient of first term and last term (3) x ( -2) = -6 Find the factors of -6 to get the middle term 5 Factors ---> +6 and -1 Write polynomial using these factors 3x2+ 6x - 1x -2 Find the common factors from 1st two terms and last two terms 3x(x +2)-1(x+2) Take common parenthesis as a common factor Factors are ( 3x-1)(x+2)

So the total factors of the given polynomial are

(x- 3)( 3x-1)(x+2)

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2) Find all possible rational x-intercepts of x
4 + 2x 3 – 7x 2 – 8x + 12.

Solution :
Constant term = 12 ⇒ Factors of 12 = p = 1,2,3,4,6,12

Leading coefficient = q = 1

Possible factors are p/q = ~+mn~ 1,2,3,4,6,12

= –12, –6, –4, –3, –2, –1, 1, 2, 3, 4, 6, 12

Polynomial

Degree of the Polynomial
Zeros of Polynomial
Remainder Theorem
Find remainder by Synthetic Division
Rational root test in Polynomial
Solved Examples on Polynomial