Using factor theorem,factorize 2x^3 -3x^2 -7x+30.
Answers
Answer:
2x3+x2-61x+30=0
Three solutions were found :
x = 5
x = -6
x = 1/2 = 0.500
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((2x3 + x2) - 61x) + 30 = 0
Step 2 :
Checking for a perfect cube :
2.1 2x3+x2-61x+30 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: 2x3+x2-61x+30
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -61x+30
Group 2: 2x3+x2
Pull out from each group separately :
Group 1: (-61x+30) • (1) = (61x-30) • (-1)
Group 2: (2x+1) • (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) = 2x3+x2-61x+30
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 2 and the Trailing Constant is 30.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,3 ,5 ,6 ,10 ,15 ,30
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 90.00
-1 2 -0.50 60.50
-2 1 -2.00 140.00
-3 1 -3.00 168.00
-3 2 -1.50 117.00
-5 1 -5.00 110.00
-5 2 -2.50 157.50
-6 1 -6.00 0.00 x+6
-10 1 -10.00 -1260.00
-15 1 -15.00 -5580.00
-15 2 -7.50 -300.00
-30 1 -30.00 -51240.00
1 1 1.00 -28.00
1 2 0.50 0.00 2x-1
2 1 2.00 -72.00
3 1 3.00 -90.00
3 2 1.50 -52.50
5 1 5.00 0.00 x-5
5 2 2.50 -85.00
6 1 6.00 132.00
10 1 10.00 1520.00
15 1 15.00 6090.00
15 2 7.50 472.50
30 1 30.00 53100.00
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
2x3+x2-61x+30
can be divided by 3 different polynomials,including by x-5
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : 2x3+x2-61x+30
("Dividend")
By : x-5 ("Divisor")
dividend 2x3 + x2 - 61x + 30
- divisor * 2x2 2x3 - 10x2
remainder 11x2 - 61x + 30
- divisor * 11x1 11x2 - 55x
remainder - 6x + 30
- divisor * -6x0 - 6x + 30
remainder 0
Quotient : 2x2+11x-6 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring 2x2+11x-6
The first term is, 2x2 its coefficient is 2 .
The middle term is, +11x its coefficient is 11 .
The last term, "the constant", is -6
Step-1 : Multiply the coefficient of the first term by the constant 2 • -6 = -12
Step-2 : Find two factors of -12 whose sum equals the coefficient of the middle term, which is 11 .
-12 + 1 = -11
-6 + 2 = -4
-4 + 3 = -1
-3 + 4 = 1
-2 + 6 = 4
-1 + 12 = 11 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 12
2x2 - 1x + 12x - 6
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (2x-1)
Add up the last 2 terms, pulling out common factors :
6 • (2x-1)
Step-5 : Add up the four terms of step 4 :
(x+6) • (2x-1)
Which is the desired factorization
Equation at the end of step 2 :
(2x - 1) • (x + 6) • (x - 5) = 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.
Answer:
Factorize: 2x3 – 3x2
-17x +30