Multiplying Polynomials #3 - Larger Polynomials
A polynomial is an algebraic expression that is comprised of two or more individual terms. In order to multiply two polynomials, you must go through a series of multiplications involving the individual terms, then add up all the results to get a final answer.
Polynomial multiplication works much like multiplying larger numbers. The problem 213 × 112 is done below. Notice that since there are three digits the first number and three digits in the second number, there are 3 × 3 = 9 individual multiplications that must take place in order to do the problem.
One common application of multiplication is in finding area. In fact, many mathematical thinkers use an area model to perform multiplications. The problem 213 × 112 can be done as a sum of areas by separating each digit.
Notice that each result in the area multiplication corresponds to an answer in the vertical multiplication. For example, the 400-20-6 on the last line of the area multiplication corresponds to the 426 which is located on the first line of the vertical multiplication. Each result equals four hundred twenty-six and comes from the multiplication of 213 × 2.
The smallest kind of polynomial has two terms and is called a binomial. When multiplying a pair of binomials, the method most often used is called FOIL, which stands for First, Outside, Inside, and Last. However, two binomials can also be done using an area model for multiplication. This method is called the “box” method because to do it you can simply draw a box. Observe how the problem (x + 3)(x + 5) is done using each method.
If you study the two answers above, you will find that they were done the exact same way mathematically. The only difference was the thought process used to get to the answer. When dealing with binomials, the FOIL method is a generally the better option because there are only four separate multiplications to be done. However, when dealing with larger polynomials it is much easier to forget a multiplication or to make a mistake. The box method is the option that gives you the best chance to get the right answer. Two trinomials are multiplied below using the box method.
Each of the nine boxes represents an individual multiplication. To multiply the polynomials correctly, you must simply multiply each term of the first polynomial by each term of the second polynomial. The box method is an organizational tool that ensures you do each of the required multiplications. Take another look at the box method from the problem above. In this problem, the diagonals of the box contain all the like terms. The diagram below demonstrates how the like terms are all contained within a diagonal. It is easy to see where the 9b3, 25b2, and 30b come from when looking at it this way.
The like terms are along the diagonals in many of the multiplications in this lesson. In fact, any multiplication problem where the exponents decrease by 1 in each term of the polynomials will result in a box whose diagonals contain like terms. An example of this is shown below.
The box that you must draw will change shapes according to the size of the polynomials. As the polynomials get larger, box must increase in size accordingly. The following problem has a four term polynomial and a five term polynomial. Therefore the box that must be drawn is 4 × 5.
Once you complete the box, you still have to collect like terms and add them to get the final answer. This can be a tedious process and it is easy to leave out one or more of the terms as you are doing the adding. For this problem, a fast way to identify like terms is to look along the diagonals.
See if you can do this final problem by yourself. The best way to test your understanding is to try to do the entire problem, then reveal the answer below to you see if you got it right.
Multiply each pair of expressions:
1) (2f + 4)(3f + 7)
2) (g2 + 3g + 5)(5g – 8)
3) -3h (h4 + 9h3 – 6h2 – 14h + 9)
4) (j3 + 6j2 + 4j + 3)(2j2 + 3j + 7)
5) (2k3 + 4k2 – 6k + 1)(k4 + 3k3 – 10k2 – k + 2)
Scroll down for answers
1) 6f2 + 26f + 28
2) 5g3 + 7g2 + g – 40
3) -3h5 – 27h4 + 18h3 + 42h2 – 27h
4) 2j5 + 15j4 + 33j3 + 60j2 + 37j + 21
5) 2k7 + 10k6 – 14k5 – 59k4 + 63k3 + 4k2 – 13k + 2
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