Operations with Polynomials – Tutorial and Practice

When we are adding or subtracting 2 or more polynomials, we have to first group the same variables (arguments) that have the same degrees and then add or subtract them. For example, if we have ax³ in one polynomial (where a is some real number), we have to group it with bx³ from the other polynomial (where b is also some real number).  Here is one example with adding polynomials:</p>

=(-x²+2x+3)+(2x²+4x-5)=

-x²+2x+3+2x²+4x-5=

x²+6x-2

We remove the brackets, and since we have a plus in front of every bracket, the signs in the polynomials don’t change.

We group variables with the same degrees: red is for second degree, and there we have -1+2, which is 1 and that’s how we got x². Blue is for the first degree where we have 2+4 which is 6, and the green is for the constants (real numbers) where we have 3-5 which is -2.

The principle is the same with subtracting, only we have to keep in mind that a minus in front of the polynomial changes all signs in that polynomial. Here is one example:

(4x³-x²+3)-(-3x²-10)=

4x³-x²+3+3x²+10=

4x³+2x²+13

<p>We remove the brackets, and since wtation”>e have a minus in front of the second polynomial, all signs in that polynomial change. We have -3x²and with minus in front, it becomes a plus and same goes for -10 (red pluses).

Now we group the variables with same degrees: there is no variable with the third degree in the second polynomial, so we just write 4x³. We group other variables the same way when we were adding polynomials.

1. Add polynomials -3x²+2x+6 and -x²-x-1.

a. -2x²+x+5
         b. -4x²+x+5
         c. -2x²+3x+5
         d. -4x²+3x+5

2. Subtract polynomials 4x³-2x²-10 and 5x³+x²+x+5.

a. -x³-3x²-x-15
         b. 9x³-3x²-x-15
         c. -x³-x²+x-5
         d. 9x³-x²+x+5

3. Divide  x³-3x²+3x-1  by  x-1.

a. x²-1
           b. x²+1
           c. x²-2x+1
           d. x²+2x+1

4. Divide  x²-y²  by  x-y.

a. x-y
          b. x+y
          c. xy
          d. y-x

1.  B
 -4x²+x+5
(-3x²+2x+6) + (-x²-x-1)=
-3x²+2x+6 -x²-x-1=
-4x²+x+5

We remove the brackets and we group the variables by degrees.

 2. A
-x³-3x²-x-15
(4x³-2x²-10)-(5x³+x²+x+5)=
4x³-2x²-10-5x³-x²-x-5=
-x³-3x²-x-15

We remove the brackets, but we change all signs in the second polynomial because of the minus. Now we group the variables by degrees.

3.  C
x²-2x+1
(x³-3x²+3x-1) : (x-1)= x²-2x+1
-(x³-x²)
      -2x²+3x-1
   -(-2x²+2x)
                 x-1
             -(x-1)
                    0

4. B
x+y

           (x²-y²) : (x-y) =x+y
         -(x²-xy)
               xy-y²
            -(xy-y²)
                 0