Multiplying polynomials
Multiplying polynomials with the following two good examples will help you master this topic once and for all. That is all you will need. It could be very useful though to review
the
multiplication of binomials
Example #1:
Multiply 4x
^{3} + 2x + 5 by 3x
^{4} + x + 6
(4x
^{3} + 2x + 5) × (3x
^{4} + x + 6)
Important concept
You must know what a term is when multiplying polynomials. It is because the goal is to multiply each term of the polynomial on the left by each term of the polynomial on the right and then adding the whole thing!
We will show the result of each multiplication or whatever will be added together in bold
The polynomial on the left is 4x
^{3} + 2x + 5
Look at it carefuly. Each term is separated by an addition sign.
So the first term is 4x
^{3}
The second term is 2x
The third term is 5
The polynomial on the right is 3x
^{4} + x + 6
Again, each term is separated by an addition sign.
So the first term is 3x
^{4}
The second term is x
The third term is 6
Now multiply the first term of the polynomial on the left that is 4x
^{3} by each term of the polynomial on the right and these are 3x
^{4},
x, and 6.
Let's do it!
4x
^{3} × 3x
^{4} = 4 × 3 × x
^{3} × x
^{4} = 12x
^{ 3 + 4} =
12x^{7}
4x
^{3} × x = 4 × x
^{3} × x = 4 × x
^{3} × x
^{1} = 4x
^{ 3 + 1} =
4x^{4}
4x
^{3} × 6 = 4 × 6x
^{3} =
24x^{3}
Next, multiply the second term of the polynomial on the left that is 2x by each term of the polynomial on the right and these are 3x
^{4},
x, and 6.
2x × 3x
^{4} = 2 × 3 × x × x
^{4} = 2 × 3 × x
^{1} × x
^{4} = 6x
^{ 1 + 4} =
6x^{5}
2x × x = 2 × x × x = 2 × x
^{1} × x
^{1} = 2x
^{ 1 + 1} =
2x^{2}
2x × 6 = 2 × 6x =
12x
Finally, multiply the third term of the polynomial on the left that is 5 by each term of the polynomial on the right and these are 3x
^{4},
x, and 6.
5 × 3x
^{4} =
15x^{4}
5 × x =
5x
5 × 6 =
30
Adding the result in bold together, we get:
12x
^{7} + 4x
^{4} + 24x
^{3} + 6x
^{5} + 2x
^{2} + 12x + 15x
^{4} + 5x + 30
Combine like terms
12x
^{7} + (4x
^{4} + 15x
^{4}) + 24x
^{3} + 6x
^{5} + 2x
^{2} + (12x + 5x) + 30
12x
^{7} + 19x
^{4} + 24x
^{3} + 6x
^{5} + 2x
^{2} + 17x + 30
Example #2:
Multiply 4x
^{3} − 2x + 5 by 3x
^{4} + x − 6
Almost the same problem. We just incorporated a couple of subtraction signs.
My teaching experience has taught me that when multiplying polynomials, it is best to say to students to replace minus with + -
Then, do the exact same thing you did in example #1
(4x
^{3} − 2x + 5) × (3x
^{4} + x − 6) = (4x
^{3} + -2x + 5) × (3x
^{4} + x + -6)
4x
^{3} × 3x
^{4} =
12x^{7}
4x
^{3} × x =
4x^{4}
4x
^{3} × -6 = 4 × -6x
^{3} =
-24x^{3}
Notice this time that the second term of the polynomial on the left has a negative next to it! same thing for the third term of the polynomial on the right
Pay careful attention to this when multiplying polynomials!
-2x × 3x
^{4} = -2 × 3 × x × x
^{4} = -2 × 3 × x
^{1} × x
^{4} = -6x
^{ 1 + 4} =
-6x^{5}
-2x × x = -2 × x × x = -2 × x
^{1} × x
^{1} = -2x
^{ 1 + 1} =
-2x^{2}
-2x × -6 = -2 × -6x =
12x
5 × 3x
^{4} =
15x^{4}
5 × x =
5x
5 × -6 =
-30
Adding the result in bold together, we get:
12x
^{7} + 4x
^{4} + -24x
^{3} + -6x
^{5} + -2x
^{2} + 12x + 15x
^{4} + 5x + -30
Combine like terms
12x
^{7} + (4x
^{4} + 15x
^{4}) + -24x
^{3} + -6x
^{5} + -2x
^{2} + (12x + 5x) + -30
12x
^{7} + 19x
^{4} + -24x
^{3} + -6x
^{5} + -2x
^{2} + 17x + -30
Multiplying polynomials should be a breeze if you really undersdand the two examples above.