Given that (x-2)2 is a factor of x3 -x2 -8x +12 . Find the other factors.
Answers
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 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
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
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-4
Polynomial Long Division :
1.2 Polynomial Long Division
Dividing : x3-13x-12
("Dividend")
By : x-4 ("Divisor")
dividend x3 - 13x - 12
- divisor * x2 x3 - 4x2
remainder 4x2 - 13x - 12
- divisor * 4x1 4x2 - 16x
remainder 3x - 12
- divisor * 3x0 3x - 12
remainder 0
Quotient : x2+4x+3 Remainder: 0
Trying to factor by splitting the middle term
1.3 Factoring x2+4x+3
The first term is, x2 its coefficient is 1 .
The middle term is, +4x its coefficient is 4 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant 1 • 3 = 3
Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is 4 .
-3 + -1 = -4
-1 + -3 = -4
1 + 3 = 4 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 3
x2 + 1x + 3x + 3
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 :
3 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+3) • (x+1)
Which is the desired factorization
Final result :
(x + 3) • (x + 1) • (x - 4)
Step-by-step explanation:
Answer:
Here one root is x = 1 so (x - 1) is a factor. i.e
{x}^{3} - {x}^{2} - 3 {x}^{2} + 3x + 2x - 2 =x
3
−x
2
−3x
2
+3x+2x−2=
{x}^{2} (x - 1) - 3x(x - 1) + 2(x - 1) =x
2
(x−1)−3x(x−1)+2(x−1)=
(x - 1)( {x}^{2} - 3x + 2) =(x−1)(x
2
−3x+2)=
(x - 1)( {x}^{2} - 2x - x + 2) =(x−1)(x
2
−2x−x+2)=
(x - 1)(x(x - 2) - 1(x - 2)) =(x−1)(x(x−2)−1(x−2))=
(x - 1)(x - 2)(x - 1) = (x - 1) {}^{2} (x - 2)(x−1)(x−2)(x−1)=(x−1)
2
(x−2)