Perform the denoted operation for each of the following.

(a) Add 6x^{5} -10x^{2} + x - 45 to 13x^{2} - 9 x + 4 .

(b) Subtract 5x^{3} - 9 x^{2} + x - 3 from x^{2+} x +1.

**Solution**

(a) Add 6x^{5} -10x^{2} + x - 45 to 13x^{2} - 9 x + 4 .

The first thing which we have to do is in fact write down the operation which we are being asked to do.

(6 x^{5} -10 x^{2} + x - 45) +(13x^{2} - 9 x + 4)

In this case the parenthesis is not needed since we are going to add the two polynomials. They are there basically to make clear the operation which we are performing. In order to add two polynomials all that we do is combine such as terms. It means that for each term with the similar exponent we will add or subtract the coefficient of that term.

In this case this is,

(6x^{5} -10x^{2} + x - 45) ^{+} (13x^{2} - 9 x + 4) =6 x^{5} + (-10 + 13) x^{2} + (1 - 9) x - 45 + 4

= 6x^{5} + 3x^{2} - 8x - 41

(b) Subtract 5x^{3} - 9 x^{2} + x - 3 from x^{2} + x + 1.

Again, let's write down the operation we are doing here. We will also need to be very careful with the order that we write things down in. Here is the operation

x^{2} + x + 1 - (5x^{3} - 9 x^{2} + x - 3)

This time the parentheses about the second term are absolutely needed. We are subtracting the whole polynomial & the parenthesis has to be there to ensure we are actually subtracting the whole polynomial.

In performing the subtraction the first thing which we'll do is distribute the minus sign through the parenthesis. It means that we will alter the sign on every term into the second polynomial. Notice that all we are actually doing here is multiplying a "-1" to the second polynomial via the distributive law. After distributing the minus through the parenthesis again we combine like terms.

Here is the work for this problem.

x^{2} + x + 1 - (5x^{3 } - 9 x2 +x - 3) = x^{2} + x + 1 - 5x^{3} + 9 x^{2} - x + 3

= -5x^{3} + 10x^{2} + 4

Notice that sometimes a term will totally drop out after combing such as terms as the x did here. It will happen on occasion thus don't get excited about it while it does happen.

Now let's move over multiplying polynomials. Again, it's best to do these in an instance.