Factorise: 5 − (3x² − 2x) (6 − 3x² + 2x).
Answers
Answer:
5-(3x2-2x)(6-3x2+2x)
Final result :
(3x + 1) • (x + 1) • (x - 1) • (3x - 5)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
5-(((3•(x2))-2x)•((6-3x2)+2x))
Step 2 :
Equation at the end of step 2 :
5 - ((3x2 - 2x) • (-3x2 + 2x + 6))
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
3x2 - 2x = x • (3x - 2)
Trying to factor by splitting the middle term
4.2 Factoring -3x2 + 2x + 6
The first term is, -3x2 its coefficient is -3 .
The middle term is, +2x its coefficient is 2 .
The last term, "the constant", is +6
Step-1 : Multiply the coefficient of the first term by the constant -3 • 6 = -18
Step-2 : Find two factors of -18 whose sum equals the coefficient of the middle term, which is 2 .
-18 + 1 = -17
-9 + 2 = -7
-6 + 3 = -3
-3 + 6 = 3
-2 + 9 = 7
-1 + 18 = 17
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 4 :
5 - x • (3x - 2) • (-3x2 + 2x + 6)
Step 5 :
Polynomial Roots Calculator :
5.1 Find roots (zeroes) of : F(x) = 9x4-12x3-14x2+12x+5
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 9 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1,3 ,9
of the Trailing Constant : 1 ,5
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x+1
-1 3 -0.33 0.00 3x+1
-1 9 -0.11 3.51
-5 1 -5.00 6720.00
-5 3 -1.67 71.11
-5 9 -0.56 -3.07
1 1 1.00 0.00 x-1
1 3 0.33 7.11
1 9 0.11 6.15
5 1 5.00 3840.00
5 3 1.67 0.00 3x-5
5 9 0.56 6.15
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
9x4-12x3-14x2+12x+5
can be divided by 4 different polynomials,including by 3x-5
Polynomial Long Division :
5.2 Polynomial Long Division
Dividing : 9x4-12x3-14x2+12x+5
("Dividend")
By : 3x-5 ("Divisor")
dividend 9x4 - 12x3 - 14x2 + 12x + 5
- divisor * 3x3 9x4 - 15x3
remainder 3x3 - 14x2 + 12x + 5
- divisor * x2 3x3 - 5x2
remainder - 9x2 + 12x + 5
- divisor * -3x1 - 9x2 + 15x
remainder - 3x + 5
- divisor * -x0 - 3x + 5
remainder 0
Quotient : 3x3+x2-3x-1 Remainder: 0
Polynomial Roots Calculator :
5.3 Find roots (zeroes) of : F(x) = 3x3+x2-3x-1
See theory in step 5.1
In this case, the Leading Coefficient is 3 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x+1
-1 3 -0.33 0.00 3x+1
1 1 1.00 0.00 x-1
1 3 0.33 -1.78
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
3x3+x2-3x-1
can be divided by 3 different polynomials,including by x-1
Polynomial Long Division :
5.4 Polynomial Long Division
Dividing : 3x3+x2-3x-1
("Dividend")
By : x-1 ("Divisor")
dividend 3x3 + x2 - 3x - 1
- divisor * 3x2 3x3 - 3x2
remainder 4x2 - 3x - 1
- divisor * 4x1 4x2 - 4x
remainder x - 1
- divisor * x0 x - 1
remainder 0
Quotient : 3x2+4x+1 Remainder: 0
Trying to factor by splitting the middle term
5.5 Factoring 3x2+4x+1
The first term is, 3x2 its coefficient is 3 .
The middle term is, +4x its coefficient is 4 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 3 • 1 = 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
3x2 + 1x + 3x + 1
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (3x+1)
Add up the last 2 terms, pulling out common factors :
1 • (3x+1)
Step-5 : Add up the four terms of step 4 :
(x+1) • (3x+1)
Which is the desired factorization