f(x)=x^3-x^2-12x+32=2
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x3-x2-12x=-32
Three solutions were found :
x = -4
x =(5-√-7)/2=(5-i√ 7 )/2= 2.5000-1.3229i
x =(5+√-7)/2=(5+i√ 7 )/2= 2.5000+1.3229i
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
x^3-x^2-12*x-(-32)=0
Step by step solution :
Step 1 :
Checking for a perfect cube :
1.1 x3-x2-12x+32 is not a perfect cube
Trying to factor by pulling out :
1.2 Factoring: x3-x2-12x+32
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-x2
Group 2: -12x+32
Pull out from each group separately :
Group 1: (x-1) • (x2)
Group 2: (3x-8) • (-4)
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 :
1.3 Find roots (zeroes) of : F(x) = x3-x2-12x+32
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 32.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8 ,16 ,32
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 42.00 -2 1 -2.00 44.00 -4 1 -4.00 0.00 x+4 -8 1 -8.00 -448.00 -16 1 -16.00 -4128.00 -32 1 -32.00 -33376.00 1 1 1.00 20.00 2 1 2.00 12.00 4 1 4.00 32.00 8 1 8.00 384.00 16 1 16.00 3680.00 32 1 32.00 31392.00
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-x2-12x+32
can be divided with x+4
Polynomial Long Division :
1.4 Polynomial Long Division
Dividing : x3-x2-12x+32
("Dividend")
By : x+4 ("Divisor")
dividend x3 - x2 - 12x + 32 - divisor * x2 x3 + 4x2 remainder - 5x2 - 12x + 32 - divisor * -5x1 - 5x2 - 20x remainder 8x + 32 - divisor * 8x0 8x + 32 remainder 0
Quotient : x2-5x+8 Remainder: 0
Trying to factor by splitting the middle term
1.5 Factoring x2-5x+8
The first term is, x2 its coefficient is 1 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is +8
Step-1 : Multiply the coefficient of the first term by the constant 1 • 8 = 8
Step-2 : Find two factors of 8 whose sum equals the coefficient of the middle term, which is -5 .
-8 + -1 = -9 -4 + -2 = -6 -2 + -4 = -6 -1 + -8 = -9 1 + 8 = 9 2 + 4 = 6 4 + 2 = 6 8 + 1 = 9
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 1 :
(x2 - 5x + 8) • (x + 4) = 0
x3-x2-12x=-32
Three solutions were found :
x = -4
x =(5-√-7)/2=(5-i√ 7 )/2= 2.5000-1.3229i
x =(5+√-7)/2=(5+i√ 7 )/2= 2.5000+1.3229i
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
x^3-x^2-12*x-(-32)=0
Step by step solution :
Step 1 :
Checking for a perfect cube :
1.1 x3-x2-12x+32 is not a perfect cube
Trying to factor by pulling out :
1.2 Factoring: x3-x2-12x+32
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-x2
Group 2: -12x+32
Pull out from each group separately :
Group 1: (x-1) • (x2)
Group 2: (3x-8) • (-4)
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 :
1.3 Find roots (zeroes) of : F(x) = x3-x2-12x+32
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 32.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8 ,16 ,32
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 42.00 -2 1 -2.00 44.00 -4 1 -4.00 0.00 x+4 -8 1 -8.00 -448.00 -16 1 -16.00 -4128.00 -32 1 -32.00 -33376.00 1 1 1.00 20.00 2 1 2.00 12.00 4 1 4.00 32.00 8 1 8.00 384.00 16 1 16.00 3680.00 32 1 32.00 31392.00
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-x2-12x+32
can be divided with x+4
Polynomial Long Division :
1.4 Polynomial Long Division
Dividing : x3-x2-12x+32
("Dividend")
By : x+4 ("Divisor")
dividend x3 - x2 - 12x + 32 - divisor * x2 x3 + 4x2 remainder - 5x2 - 12x + 32 - divisor * -5x1 - 5x2 - 20x remainder 8x + 32 - divisor * 8x0 8x + 32 remainder 0
Quotient : x2-5x+8 Remainder: 0
Trying to factor by splitting the middle term
1.5 Factoring x2-5x+8
The first term is, x2 its coefficient is 1 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is +8
Step-1 : Multiply the coefficient of the first term by the constant 1 • 8 = 8
Step-2 : Find two factors of 8 whose sum equals the coefficient of the middle term, which is -5 .
-8 + -1 = -9 -4 + -2 = -6 -2 + -4 = -6 -1 + -8 = -9 1 + 8 = 9 2 + 4 = 6 4 + 2 = 6 8 + 1 = 9
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 1 :
(x2 - 5x + 8) • (x + 4) = 0
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