Question

find the zeroes of y^3 - 3y^2 + 4y + 2.

Answer 1

y 3 -3y 2 -4y=0
Three solutions were found :
1. y = 4
2. y = -1
3. y = 0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((y 3 ) - 3y 2 ) - 4y = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
y 3 - 3y 2 - 4y = y • (y 2 - 3y - 4)
Trying to factor by splitting the middle term
3.2 Factoring y 2 - 3y - 4
The first term is, y 2 its coefficient is 1 .
The middle term is, -3y its coefficient is
-3 .
The last term, "the constant", is -4
Step-1 : Multiply the coefficient of the first term by the constant 1 • -4 = -4
Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is -3 .
-4 + 1 = -3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and 1
y 2 - 4y + 1y - 4
Step-4 : Add up the first 2 terms, pulling out like factors :
y • (y-4)
Add up the last 2 terms, pulling out common factors :
1 • (y-4)
Step-5 : Add up the four terms of step 4 :
(y+1) • (y-4)
Which is the desired factorization
Equation at the end of step 3 :
y • (y + 1) • (y - 4) = 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 : y = 0
Solution is y = 0
Solving a Single Variable Equation :
4.3 Solve : y+1 = 0
Subtract 1 from both sides of the equation :
y = -1
Solving a Single Variable Equation :
4.4 Solve : y-4 = 0
Add 4 to both sides of the equation :