factorize 5x^3-15x^2-x+3
Answers
Answer:
5x3-15x2-x+3=0
Three solutions were found :
x = 3
x = ±√ 0.200 = ± 0.44721
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((5 • (x3)) - (3•5x2)) - x) + 3 = 0
Step 2 :
Equation at the end of step 2 :
((5x3 - (3•5x2)) - x) + 3 = 0
Step 3 :
Checking for a perfect cube :
3.1 5x3-15x2-x+3 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 5x3-15x2-x+3
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -x+3
Group 2: 5x3-15x2
Pull out from each group separately :
Group 1: (-x+3) • (1) = (x-3) • (-1)
Group 2: (x-3) • (5x2)
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Add up the two groups :
(x-3) • (5x2-1)
Which is the desired factorization
Trying to factor as a Difference of Squares :
3.3 Factoring: 5x2-1
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 5 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Equation at the end of step 3 :
(5x2 - 1) • (x - 3) = 0
Step 4 :
Theory - Roots of a product :
4.1 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 :
4.2 Solve : 5x2-1 = 0
Add 1 to both sides of the equation :
5x2 = 1
Divide both sides of the equation by 5:
x2 = 1/5 = 0.200
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 1/5
The equation has two real solutions
These solutions are x = ±√ 0.200 = ± 0.44721
Solving a Single Variable Equation :
4.3 Solve : x-3 = 0
Add 3 to both sides of the equation :
x = 3
Three solutions were found :
x = 3
x = ±√ 0.200 = ± 0.44721
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