Rational root test in Polynomial
Rational root test in Polynomiala) 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
Leading coefficient = 3
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 | 3x^{2} + 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 | 3x^{2}+ 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