Multiplication of Polynomials
Transcript Start
This lesson covers the multiplication of polynomials.
First, we will begin by multiplying a polynomial by a monomial.
We will use the distributive property and the law of exponents to multiply a monomial times a binomial. We will, first, multiply the expression on the outside of the parentheses by both terms inside the parentheses using the distributive properties.
4x times the quantity, 2x plus 3, end quantity; equals 4x times 2x, plus 4x times 3; equals 8x squared plus 12x, using the law of exponents.
Let us work another problem that is a bit longer; a monomial times a polynomial with four terms.
5x times the quantity, 6y to the fourth power, minus 3x squared, minus 7y, minus 4, end quantity; equals 5x times 6y to the fourth power, plus 5x times negative 3x squared, plus 5x times negative 7y, plus 5x times negative 4; equals 30xy to the fourth power, minus 15x cubed, minus 35xy, minus 20x.
You would use the same method for a student with low vision, but on the computer, you would use a dark, uncluttered font of sufficient size for the student’s needs.
When you or the student write out the steps, you would write sufficiently large, and use a black marker such as a Sharpie, 20-20, or Flair pen on white, or canary yellow paper. Or use a combination of color and contrast of the student’s choice.
For a student who is blind, all these steps in print can be written in braille in the student’s choice of math codes on a braillewriter or on a braille notetaker.
Here is what the first problem of a monomial times a binomial would look like in the Nemeth Code.
Second, we will be multiplying two binomials.
We can use the distributive property twice to multiply two binomials. Let’s see what that looks like for the problem.
The quantity x plus 4, end quantity, times the quantity x plus 5, end quantity; equals x times the quantity x plus 5, end quantity, plus 4 times the quantity x plus 5, end quantity; equals x times x, plus x times 5, plus 4 times x, plus 4 times 5; equals x squared, plus 5x, plus 4x, plus 20; equals x squared, plus nine x, plus 20.
The mnemonic FOIL, F-O-I-L, First, Outside, Inside, Last method can be used in English-speaking countries to shorten this process. Other countries may have a similar short-cut method.
The box or grid method could also be used.
So, place one binomial, x+4, at the top of the grid, place x+5 on the left of the grid, and make sure the terms line up with a row or column. Now we're going to multiply the rows and columns, of the grid, to complete the interior of the grid.
X times x is x squared.
X times 4 is 4x.
5 times x is 5x.
5 times 4 is 20.
And then we add all the boxes together and we get,
x squared plus-- 5x plus 4x makes 9x, plus 20.
We can even use columns to multiply. That is, the binomials are written one above the other. We multiply each term on the top row by each term on the bottom, and then we add.
X times x is x squared.
X times 4 is 4x.
5 times x is 5x.
5 times 4 is 20.
And then down our columns,
x squared-- 4x plus 5x is 9x, and then, plus 20.
There is yet another method using algebra tiles to illustrate the multiplication of these two binomials.
You would use the same method for a student with low vision, but on the computer, you would use a dark uncluttered font of sufficient size for the student’s needs.
When you or the student write out the steps, you would write sufficiently large, and use a black marker such as a Sharpie, 20-20, or Flair pen on white or canary yellow paper. Or use a combination of color and contrast of the student’s choice.
For a student who is blind, all these steps in print can be written in braille in the student’s choice of math codes on a braillewriter or on a braille notetaker.
The Algebra Tiles would need to be adapted. For example, texture could be substituted for color. You can also make your own. The majority of my students prefer a quick, orderly method, such as the FOIL method, so that that they can do mental math for the majority of the problem, writing answers in braille, as they go; and then adding the like terms and placing in descending order for their final answer.
Nevertheless, if the student is still having difficulty, you could use a spatial mathematical board. The Cubarithm board could be used for the detached coefficient method, shown here. Or the MathWindow could be used for both the full spatial arrangement, and the detached coefficient method.
Finally, we will be multiplying any two polynomials.
To multiply two polynomials, multiply each term of one polynomial by every term of the other. Add the results.
We will illustrate this procedure by multiplying a binomial by a trinomial. We will write the answer in descending order of the variable.
The quantity, 4x minus 3, end quantity, times the quantity, x squared, plus x, plus 5; equals 4x cubed, plus 4x squared, plus 20x, minus 3x squared, minus 3x, minus 15; equals 4x cubed, plus x squared, plus 17x, minus 15.
The box or grid method could also be used with this problem.
The spatial method of columns could be used, as well.
We have already discussed the adaptations necessary to make all of these methods, materials accessible for a student with a visual impairment.
In this video we have covered multiplying a polynomial by a monomial, multiplying two binomials, and multiplying any two polynomials.