Factoring Polynomials

Question

Factor and check
–c^4-c^3+56c^2

Answer 1

The highest shred factor of c is c^2.  This can be factored out

 

c^2(-c^2-c+56)

 

The term in brackets can be factored with the following reasoning.

Let x,y be real numbers.  (-c+x)(c+y)=-c^2 +(x-y)c +xy

We can therefore choose xy=56 to match the term in brackets

7*8 =56  and 7-8=-1 are obvious tries which match the coefficients

 

So we can write

 

–c^4-c^3+56c^2=c^2(-c+7)(c+8) which is nicer to write as c^2(7-c)(c+8)

 

I will  leave it  to you to show a check

Answer 2

-c⁴ - c³ + 56c²

 

To factor, we take out the GCF.  The GCF is c².

 

c²(-c² - c + 56)

 

or

 

-c²(c² + c - 56)  ------->  This form is useful when solving a quadratic equation or an equation with a degree higher than 2.

 

For instance:

c² + c - 56  can be factored out as  (c - 7)(c + 8).

 

To check, simply distribute the GCF into each term in parentheses.

 

 

Answer 3

–c⁴ - c³ + 56c²

 

Factor out a -c²:

 

-c²(c² + c - 56)

 

The quadratic in the parentheses factor to (c+8)(c-7):

 

-c²(c+8)(c-7)