Similar to the operations of addition and subtraction of polynomials, the operation of multiplication on polynomials can also be understood through the basic concepts of number multiplication and the distributive property.
The main principle in multiplying two polynomials is: multiply each term in the first polynomial by each term in the second polynomial.
After performing all the multiplications between terms, the next step is to combine (add or subtract) like terms to simplify the result.
This method organizes the multiplication of each term using a table, similar to finding the area when multiplying two numbers.
For example, the multiplication 16×12 can be seen as the area of a rectangle with sides 10+6 and 10+2.
10
6
10
100
60
2
20
12
Total area is 100+60+20+12=192.
The same approach can be applied to polynomials.
Example3:
Find the product of (x+6)(x+2) using the table method.
x
+6
x
x2
+6x
+2
+2x
+12
Now, sum all the results inside the table cells:
x2+6x+2x+12
Combine like terms:
=x2+(6x+2x)+12
=x2+8x+12
Example4:
Find the product of (x−5)(x2+3x−1) using the table method.
x2
+3x
−1
x
x3
+3x2
Sum all the results inside the table cells:
x3+3x2−x−5x2−15x+5
Combine like terms:
=x3+(3x2−5x2)+(−x−15x)+5
=x3−2x2−16x+5
Notice that the result from the table method is the same as the result from the horizontal distribution method. The table method is just another way to organize the multiplication of each term.
