MathematicsRelations and Functions: Multiplying Polynomials

Science and Mathematics Education Research Group

Supported by UBC Teaching and Learning Enhancement Fund 2012-2015

Department of Curr iculum and Pedagogy

F A C U L T Y O F E D U C A T I O Na place of mind

Factoring Polynomials II

Retrieved from http://catalog.flatworldknowledge.com/bookhub/reader/6329?e=fwk-redden-ch06_s01

Multiplying Polynomials I

Multiply completely:

5

6

5

6

5

26E.

12D.

12C.

8B.

8A.

x

x

x

x

x

Solution

Answer: C

Justification:

Multiplying polynomials is to multiply factors in a division statement into one single expression.

In order to multiply polynomials, we multiply coefficients with coefficients, and variables with variables.

For coefficients, we multiply 2 with 6, which is 12.

For variables, we multiply with , which is

()

Multiplying them back together, we get 12 Our answer is C.

Multiplying Polynomials II

Multiply completely:

zx

zyx

zyx

zx

zx

5

46

45

6

5

E.

6D.

6C.

6B.

6A.

Solution

Answer: A

Justification:

In order to multiply polynomials, we multiply coefficients with coefficients, and variables with variables

For coefficients, we multiply 2 with -3, which is -6.

For x, we multiply with , which is

For y, we multiply with , which is

For z, there is only 1 z; thus it is just z

() ()

Multiplying them back together, we get 6 Our answer is A.

Multiplying Polynomials III

Expand:

)7(E.

)12(D.

)12(C.

)12(B.

)1(A.

2

2

2

2

2

xx

xx

xx

x

x

Solution

Answer: C

Justification:

In order to multiply polynomials with more than 1 term, we have to “distribute” our multiplication to each term as such!

Multiplying each term over and under, we get:

= =

Our answer is C.

It’s like fishing!

Multiplying Polynomials IV

Expand:

)4129(E.

)4129(D.

)4129(C.

)4129(B.

)49(A.

2

2

2

2

2

xx

xx

xx

xx

x

Solution

Answer: E

Justification:

Notice that this question can also be written as a multiple of two factors as such!

Multiplying each term over and under, we get:

= =

Our answer is E.

≠

Multiplying Polynomials V

Expand:

2

23

23

22

23

3E.

)132(D.

)132(C.

)12(B.

)122(A.

x

xx

xx

xx

xxx

Solution

Answer: D

Justification:

Notice that this question cannot be written as a multiple of two factors as such!

Due to the order of operation, we must evaluate the exponents first!

= =

≠ ()

Solution Continued

Now, this question can be written as multiple of two factors as such!

Since there is a negative sign in front of it, we multiply to all terms.

Thus, our answer is D.

Multiplying Polynomials VI

Expand:

3223

3223

3223

3223

33

6E.

33D.

C.

2B.

A.

yyxx

yxyyxx

yxyyxx

yyxx

yx

Solution

Answer: D

Justification:

Notice that this question can be written as a multiple of three factors as such!

Due to the order of operation, we must evaluate the first exponents first!

Then, we multiply this result with the last factor

Thus, the answer is D.

≠ ()

Multiplying Polynomials VII

Expand:

54322345

541322345

541322345

542345

5445

510105E.

510105D.

510105C.

5105B.

55A.

yxyyxyxyxx

yyxyxyxyxx

yyxyxyxyxx

yyyxxx

yyxx

Solution

Answer: E

Justification:

We can solve this problem by using Pascal’s triangle

Retrieved from http://en.wikipedia.org/wiki/Pascal's_triangle#/media/File:Pascal%27s_triangle_5.svg

Pascal’s triangle is a triangular model that starts with 1 at the top, continues to place number below in a triangular shape, and has pattern that each number of a row is the two numbers above added together excluding the edges.

It is found that row of the Pascal’s triangle represents the coefficients of the terms in a polynomial , where n is a positive integer.

Solution Continued

Since n = 5 for our example, we know that the coefficients of the terms of are at the 6th row of the Pascal’s triangle. (1)

However, we have a negative sign between x and y:

This means that each term will have an alternating sign in a descending order from the first term. (2)

For example, when we have , we should get: . As you can see, the coefficients of these terms are +1, 2, and +1, respectively. They alternate sign as they progress.

22 2 yxyx

Solution Continued

Retrieved from http://en.wikipedia.org/wiki/Pascal's_triangle#/media/File:Pascal%27s_triangle_5.svg

Thus, incorporating both (1) and (2), our solution will look as below:

Thus, our answer is E.

54322345 510105 yxyyxyxyxx

+ Alternate signs