p(x) is x³-5x²-2x+24
Answers
Explanation:
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is 24.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-5x2-2x+24
can be divided by 3 different polynomials,including by x-4 .
Factoring x2-x-6
The first term is, x2 its coefficient is 1 .
The middle term is, -x its coefficient is -1 .
The last term, "the constant", is -6
Step-1 : Multiply the coefficient of the first term by the constant 1 • -6 = -6
Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is -1 .
-6 + 1 = -5 -3 + 2 = -1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -3 and 2
x2 - 3x + 2x - 6
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-3)
Add up the last 2 terms, pulling out common factors :
2 • (x-3)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x-3)
Which is the desired factorization.
so, x³-5x²-2x+24 = (x-4)(x+2)(x-3)
x= 4 or x= -2 or x = 3.....
Answer:-
The polynomial
x³-5x²-2x+24
and compare with
ax³+bx²+cx+d
α+β+y= -b/a =5
αβy = -d/a = -24
12y= -24....(as given the product
of.two zeros
i.e. αβ=12
y= -2.
α+β+y=5
α+β-2=5
α+β =7. .................(1)
(α+β)²=7²
(α-β)²+4αβ=49
(α-β)²+4*12=49
(α-β)²+48= 49
(α-β)² =1
α-β = √1
α-β=1. ....................(2)
subtracting (2) From (1)
α+β-(α-β)=7-1
α+β-α+β=6
2β =6
β=3
putting the value of β in the (2) eq
α-3=1
α=4
α=4,β=3,y= -2
i hope it helps you.