Arithmetic operations can be applied to polynomials much like they are applied to numbers.

It is best to think of numbers in expanded form, then see how they match polynomials.

If we express the number 435 in expanded form it looks like

4(100) + 3(10) + 5, or 4(10^{2}) + 3(10) + 5

Compare the above with 4*x*^{2} + 3*x* + 5

Now add 4*x*^{2} + 3*x* + 5 to 2*x*^{2} + 6*x* + 2.

This is like adding 435 to 262.

435

+262

697

(4*x*^{2} + 3*x* + 5) + (2*x*^{2} + 6*x* + 2) = 6*x*^{2} + 9*x* + 7, obtained by adding like terms from each polynomial.

So, do not panic, you use the same techniques you learned for arithmetic.

Actually, polynomial addition is easier, since there is no need to carry.

The numbers are not restricted to non-negative integers, negative values and fractions are allowed.

Subtraction is also the same technique as the one you know.

628 – 312 = 316.

Subtract 3*x*^{2} + *x* + 2 from 6*x*^{2} + 2*x* + 8

(6*x*^{2} + 2*x* + 8) – (3*x*^{2} + *x* + 2) = 3*x*^{2} + *x* + 6

Notice we subtract like terms from like terms.

Now multiply.

321 X 12

321

12

642

321

3852

Why is the 321 indented from the right? Well, look at how it occurs. 1(10) X 1 = 1(10), so the 1 belongs in the 10 column. 1(10) X 2(10) = 2(10^{2}), so the 2 belongs in the 10^{2} column. And 1(10) X 3(10^{2}) = 3(10^{3}), so the 3 belongs in the 10^{3} column.

Notice that 3(10) will not multiply by anything to get a value in the units column.

Notice each digit from one number multiplied by each digit from the other number.

(3*x*^{2} + 2*x* + 1)( *x* + 2) = 3*x*^{3} + 6*x*^{2} + 2*x*^{2} + 4*x* + *x* + 2

= 3*x*^{3} + 8*x*^{2} + 5*x* + 2

Now divide (4*x*^{2} + 6*x* + 5) ÷ (2*x* + 1)

If you divide 465/21 you get 21 with a remainder of 2, or 22 + 3/21.

Divide (4*x*^{2} + 7*x* + 5) ÷ (2*x* + 2) = 2*x* + 2 + 3/(2*x* + 1)

2 *x* + 2

2*x* + 1)4*x*^{2} + 6*x* + 5

4*x*^{2} + 2*x*

4*x* + 5

4x + 2

3

So, what is done with polynomials is the same thing done with arithmetic. It does allow negative numbers, fractions, and larger numbers, but the technique is the same.

Unfortunately, many students become calculator dependent, so these are new techniques to them. But to those who learned these techniques at an early age, the operations with polynomials are the same as those with which they are familiar.

## Comments

Loginblackspanielgallery on 02/12/2021I fixed it. I had changed the number so we did not get a value past 10 when adding, and left the wording.

DerdriuMarriner on 02/12/2021blackspanielgallery, Thank you for the practicalities and products.

I appreciate the explanations, since I always loved polynomials in school.

Under the multiplied number example, you ask us, "Why is the 963 indented?" I do not see that number anywhere.