solve x^3-9x^2-7x+81=0
Answers
Answer:
2.2 Factoring: x3+9x2-9x-81
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-81
Group 2: 9x2-9x
Pull out from each group separately :
Group 1: (x3-81) • (1)
Group 2: (x-1) • (9x)
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+9x2-9x-81
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 -81.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9 ,27 ,81
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -64.00
-3 1 -3.00 0.00 x+3
-9 1 -9.00 0.00 x+9
-27 1 -27.00 -12960.00
-81 1 -81.00 -471744.00
1 1 1.00 -80.00
3 1 3.00 0.00 x-3
9 1 9.00 1296.00
27 1 27.00 25920.00
81 1 81.00 589680.00
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+9x2-9x-81
can be divided by 3 different polynomials,including by x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3+9x2-9x-81
("Dividend")
By : x-3 ("Divisor")
dividend x3 + 9x2 - 9x - 81
- divisor * x2 x3 - 3x2
remainder 12x2 - 9x - 81
- divisor * 12x1 12x2 - 36x
remainder 27x - 81
- divisor * 27x0 27x - 81
remainder 0
Quotient : x2+12x+27 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2+12x+27
The first term is, x2 its coefficient is 1 .
The middle term is, +12x its coefficient is 12 .
The last term, "the constant", is +27
Step-1 : Multiply the coefficient of the first term by the constant 1 • 27 = 27
Step-2 : Find two factors of 27 whose sum equals the coefficient of the middle term, which is 12 .
-27 + -1 = -28
-9 + -3 = -12
-3 + -9 = -12
-1 + -27 = -28
1 + 27 = 28
3 + 9 = 12 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 9
x2 + 3x + 9x + 27
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 :
9 • (x+3)
Step-5 : Add up the four terms of step 4 :
(x+9) • (x+3)
Which is the desired factorization
Equation at the end of step
2
:
(x + 9) • (x + 3) • (x - 3) = 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.
Solving a Single Variable Equation:
3.2 Solve : x+9 = 0
Subtract 9 from both sides of the equation :
x = -9
Step-by-step explanation:
Hope it helps