A polynomial of the x⁴+2x³-7x²-8x+12 is said to represent the sales of grid digital, a computer company, for the last five years. The real zeroes of this polynomial signify break even points of the company.
How can you evaluate the break even points of the company by using the following?
*Descartes' rule of signs
*Rational rout theorem
*Graphical estimation of behavior
Answers
The Factor Theorem states that if P/Q is root of a polynomial
x4 - 2x3 - 7x2 + 8x + 12
can be divided by 4 different polynomials,including by x - 3
Polynomial Long Division :
4.3 Polynomial Long Division
Dividing : x4 - 2x3 - 7x2 + 8x + 12
("Dividend")
By : x - 3 ("Divisor")
dividend x4 - 2x3 - 7x2 + 8x + 12 - divisor * x3 x4 - 3x3 remainder x3 - 7x2 + 8x + 12 - divisor * x2 x3 - 3x2 remainder - 4x2 + 8x + 12 - divisor * -4x1 - 4x2 + 12x remainder - 4x + 12 - divisor * -4x0 - 4x + 12 remainder 0
Quotient : x3+x2-4x-4 Remainder: 0
Polynomial Roots Calculator :
4.4 Find roots (zeroes) of : F(x) = x3+x2-4x-4
See theory in step 4.2
In this case, the Leading Coefficient is 1 and the Trailing Constant is -4.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 0.00 x+1 -2 1 -2.00 0.00 x+2 -4 1 -4.00 -36.00 1 1 1.00 -6.00 2 1 2.00 0.00 x-2 4 1 4.00 60.00
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+x2-4x-4
can be divided by 3 different polynomials,including by x-2
Polynomial Long Division :
4.5 Polynomial Long Division
Dividing : x3+x2-4x-4
("Dividend")
By : x-2 ("Divisor")
dividend x3 + x2 - 4x - 4 - divisor * x2 x3 - 2x2 remainder 3x2 - 4x - 4 - divisor * 3x1 3x2 - 6x remainder 2x - 4 - divisor * 2x0 2x - 4 remainder 0
Quotient : x2+3x+2 Remainder: 0
Trying to factor by splitting the middle term
4.6 Factoring x2+3x+2
The first term is, x2 its coefficient is 1 .
The middle term is, +3x its coefficient is 3 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 1 • 2 = 2
Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 3 .
-2 + -1 = -3 -1 + -2 = -3 1 + 2 = 3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 2
x2 + 1x + 2x + 2
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 :
2 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x+1)
Which is the desired factorization
We can draw the bar graph by following the given steps:
Step 1.- On a graph paper, draw a horizontal line OX as x-axis and vertical line OY as y-axis.
Step 2.- Along OX, write the names of the subjects at the points that are taken at a uniform gap.
Step 3.- Choose the scale:
1 big division = 10 marks
1 small division = 1