Answer:
STEP
1
:
Equation at the end of step 1
(((x3) - 22x2) - 3x) + 2 = 0
STEP
2
:
Checking for a perfect cube
2.1 x3-4x2-3x+2 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-4x2-3x+2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -3x+2
Group 2: x3-4x2
Pull out from each group separately :
Group 1: (-3x+2) • (1) = (3x-2) • (-1)
Group 2: (x-4) • (x2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
2.3 Find roots (zeroes) of : F(x) = x3-4x2-3x+2
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 2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x+1
-2 1 -2.00 -16.00
1 1 1.00 -4.00
2 1 2.00 -12.00
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
x3-4x2-3x+2
can be divided with x+1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-4x2-3x+2
("Dividend")
By : x+1 ("Divisor")
dividend x3 - 4x2 - 3x + 2
- divisor * x2 x3 + x2
remainder - 5x2 - 3x + 2
- divisor * -5x1 - 5x2 - 5x
remainder 2x + 2
- divisor * 2x0 2x + 2
remainder 0
Quotient : x2-5x+2 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-5x+2
The first term is, x2 its coefficient is 1 .
The middle term is, -5x its coefficient is -5 .
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 -5 .
-2 + -1 = -3
-1 + -2 = -3
1 + 2 = 3
2 + 1 = 3
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
2
:
(x2 - 5x + 2) • (x + 1) = 0
STEP
3
:
Theory - Roots of a product
3.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Parabola, Finding the Vertex:
3.2 Find the Vertex of y = x2-5x+2
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 2.5000
Plugging into the parabola formula 2.5000 for x we can calculate the y -coordinate :
y = 1.0 * 2.50 * 2.50 - 5.0 * 2.50 + 2.0
or y = -4.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2-5x+2
Axis of Symmetry (dashed) {x}={ 2.50}
Vertex at {x,y} = { 2.50,-4.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = { 0.44, 0.00}
Root 2 at {x,y} = { 4.56, 0.00}
Solve Quadratic Equation by Completing The Square
Step-by-step explanation:
