The Retired Profesor simplifies Algebra: Combining Polynomials

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(10²) + 3(10) + 5

Compare the above with 4x² + 3x + 5

Now add 4x² + 3x + 5 to 2x² + 6x + 2.

This is like adding 435 to 262.

  435

+262

  697

(4x² + 3x + 5) + (2x² + 6x + 2) = 6x² + 9x + 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 3x² + x + 2 from 6x² + 2x + 8

(6x² + 2x + 8) – (3x² + x + 2) = 3x² + 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(10²), so the 2 belongs in the 10² column.  And 1(10) X 3(10²) = 3(10³), so the 3 belongs in the 10³ 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.

(3x² + 2x + 1)( x + 2) = 3x³ + 6x² + 2x² + 4x + x + 2

= 3x³ + 8x² + 5x + 2

Now divide (4x² + 6x + 5) ÷ (2x + 1)

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

Divide (4x² + 7x + 5) ÷ (2x + 2) = 2x + 2 + 3/(2x + 1)

                    2 x + 2

2x + 1)4x² + 6x + 5

            4x² + 2x

                      4x + 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.

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