Multiplying polynomials with the following examples will help you master this topic once and for all. It could be very useful though to review the multiplication of binomials. Then, study carefully the example in the figure below.
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!
The result of each multiplication or whatever will be added together is shown in bold.
The polynomial on the left is 4x^{3} + 2x + 5
Each term is separated by an addition sign.
The polynomial on the right is 3x^{4} + x + 6
Each term is separated by an addition sign.
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.
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
Example #2 is almost the same as example #1. 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.
-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
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 understand the three examples above.
Sep 30, 22 04:45 PM