Question

factorize x^4+x^3+16x+16

Answer 1

Answer:

step by step explanation---

(1): "x2"   was replaced by   "x^2".  2 more similar replacement(s).

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 (((x4) +  (x3)) -  24x2) -  16x  = 0  

Step  2  :

Step  3  :

Pulling out like terms :

3.1     Pull out like factors :

  x4 + x3 - 16x2 - 16x  =  

 x • (x3 + x2 - 16x - 16)  

Checking for a perfect cube :

3.2    x3 + x2 - 16x - 16  is not a perfect cube

Trying to factor by pulling out :

3.3      Factoring:  x3 + x2 - 16x - 16  

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  x3 + x2  

Group 2:  -16x - 16  

Pull out from each group separately :

Group 1:   (x + 1) • (x2)

Group 2:   (x + 1) • (-16)

              -------------------

Add up the two groups :

              (x + 1)  •  (x2 - 16)  

Which is the desired factorization

Trying to factor as a Difference of Squares :

3.4      Factoring:  x2 - 16  

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 : 16 is the square of 4

Check :  x2  is the square of  x1  

Factorization is :       (x + 4)  •  (x - 4)  

Equation at the end of step  3  :

 x • (x + 4) • (x - 4) • (x + 1)  = 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  :    x = 0  

 Solution is  x = 0

Solving a Single Variable Equation :

4.3      Solve  :    x+4 = 0  

Subtract  4  from both sides of the equation :  

                     x = -4

Solving a Single Variable Equation :

4.4      Solve  :    x-4 = 0  

Add  4  to both sides of the equation :  

                     x = 4

Solving a Single Variable Equation :

4.5      Solve  :    x+1 = 0  

Subtract  1  from both sides of the equation :  

                     x = -1

Four solutions were found :

x = -1

x = 4

x = -4

x = 0