find the quadratic equations from polynomial equation and find the value of x and y in the following polynomial x³-13x+12/x-1
Answers
Step-by-step explanation:
1.1 Find roots (zeroes) of : F(x) = x3-13x+12
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
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 12.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 24.00
-2 1 -2.00 30.00
-3 1 -3.00 24.00
-4 1 -4.00 0.00 x+4
-6 1 -6.00 -126.00
-12 1 -12.00 -1560.00
1 1 1.00 0.00 x-1
2 1 2.00 -6.00
3 1 3.00 0.00 x-3
4 1 4.00 24.00
6 1 6.00 150.00
12 1 12.00 1584.00
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
x3-13x+12
can be divided by 3 different polynomials,including by x-3
Polynomial Long Division :
1.2 Polynomial Long Division
Dividing : x3-13x+12
("Dividend")
By : x-3 ("Divisor")
dividend x3 - 13x + 12
- divisor * x2 x3 - 3x2
remainder 3x2 - 13x + 12
- divisor * 3x1 3x2 - 9x
remainder - 4x + 12
- divisor * -4x0 - 4x + 12
remainder 0
Quotient : x2+3x-4 Remainder: 0
Trying to factor by splitting the middle term
1.3 Factoring x2+3x-4
The first term is, x2 its coefficient is 1 .
The middle term is, +3x its coefficient is 3 .
The last term, "the constant", is -4
Step-1 : Multiply the coefficient of the first term by the constant 1 • -4 = -4
Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is 3 .
-4 + 1 = -3
-2 + 2 = 0
-1 + 4 = 3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 4
x2 - 1x + 4x - 4
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-1)
Add up the last 2 terms, pulling out common factors :
4 • (x-1)
Step-5 : Add up the four terms of step 4 :
(x+4) • (x-1)
Which is the desired factorization
Equation at the end of step
1
:
(x + 4) • (x - 1) • (x - 3) = 0
Answer:
it's 0 [ZERO]
Step-by-step explanation:
its a big sum cant type it right now