factorise x3-x2-32x+60
Answers
Answer:
Checking for a perfect cube :
1.1 x3+x2-32x-60 is not a perfect cube
Trying to factor by pulling out :
1.2 Factoring: x3+x2-32x-60
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3+x2
Group 2: -32x-60
Pull out from each group separately :
Group 1: (x+1) • (x2)
Group 2: (8x+15) • (-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+x2-32x-60
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 -60.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,5 ,6 ,10 ,12 ,15 ,20 , etc
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -28.00
-2 1 -2.00 0.00 x+2
-3 1 -3.00 18.00
-4 1 -4.00 20.00
-5 1 -5.00 0.00 x+5
-6 1 -6.00 -48.00
-10 1 -10.00 -640.00
-12 1 -12.00 -1260.00
-15 1 -15.00 -2730.00
-20 1 -20.00 -7020.00
1 1 1.00 -90.00
2 1 2.00 -112.00
3 1 3.00 -120.00
4 1 4.00 -108.00
5 1 5.00 -70.00
6 1 6.00 0.00 x-6
10 1 10.00 720.00
12 1 12.00 1428.00
15 1 15.00 3060.00
20 1 20.00 7700.00
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+x2-32x-60
can be divided by 3 different polynomials,including by x-6
Polynomial Long Division :
1.4 Polynomial Long Division
Dividing : x3+x2-32x-60
("Dividend")
By : x-6 ("Divisor")
dividend x3 + x2 - 32x - 60
- divisor * x2 x3 - 6x2
remainder 7x2 - 32x - 60
- divisor * 7x1 7x2 - 42x
remainder 10x - 60
- divisor * 10x0 10x - 60
remainder 0
Quotient : x2+7x+10 Remainder: 0
Trying to factor by splitting the middle term
1.5 Factoring x2+7x+10
The first term is, x2 its coefficient is 1 .
The middle term is, +7x its coefficient is 7 .
The last term, "the constant", is +10
Step-1 : Multiply the coefficient of the first term by the constant 1 • 10 = 10
Step-2 : Find two factors of 10 whose sum equals the coefficient of the middle term, which is 7 .
-10 + -1 = -11
-5 + -2 = -7
-2 + -5 = -7
-1 + -10 = -11
1 + 10 = 11
2 + 5 = 7 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 5
x2 + 2x + 5x + 10
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+2)
Add up the last 2 terms, pulling out common factors :
5 • (x+2)
Step-5 : Add up the four terms of step 4 :
(x+5) • (x+2)
Which is the desired factorization
Final result :
(x + 5) • (x + 2) • (x - 6)
Explanation: