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| 11 Theory slides |
| 13 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
LHS * C=RHS* C
Distribute C
Polynomial | Degree | Leading Coefficient |
---|---|---|
P(x)= a_nx^n + a_(n-1)x^(n-1) + ⋯ + a_1x + a_0 | n | a_n |
C* P(x) = C* a_nx^n + C* a_(n-1)x^(n-1) + ⋯ + C* a_1x + C* a_0 | n | C* a_n |
A = l * w From the chat and the diagram made by Kevin, the poster has to be 2 feet shorter and narrower than the wall. Since the wall is 10 feet tall, the poster will be 8 feet tall. The wall is x feet wide, so the width of the poster is x-2 feet.
A polynomial modeling the area of the poster is obtained by substituting the poster's dimensions into the formula for the area. A(x) = (x-2) * 8 ⇓ A(x) = 8x-16 Note that this polynomial gives the area of the poster depending on the width of the wall.
A(x) = 8x - 16 ⇕ A(x) = 8x^1 - 16 The greatest exponent is 1, which means that the degree of the polynomial is 1. The coefficient in front of the term with the greatest exponent is 8. Thus, the leading coefficient is 8.
x= 16
Multiply
Subtract terms
Different methods can be used to multiply two polynomials. The following three methods are based on the Distributive Property.
Distribute x^3 + 2x^2 - 3
Multiply
Commutative Property of Addition
Associative Property of Addition
Add terms
Start by drawing a table that has as many rows as there are terms in the first polynomial and that has as many columns as there are terms in the second polynomial.
Polynomial | Number of Terms |
---|---|
P(x) = x^3 + 2x^2 - 3 | 3 |
Q(x) = x^2 + 4 | 2 |
For example, a table with 3 rows and 2 columns is needed to multiply P(x) by Q(x).
Now, write each term of the first polynomial at the left of each cell of the first column. Similarly, write each term of the second polynomial above each cell of the first row.
Start by multiplying the first terms of each binomial. In this case, multiply x by 3x. ( x+6)( 3x-2) = x( 3x) The empty box is there as a reminder that there are still missing terms.
Next, multiply the outer terms — that is, multiply the first term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply x by -2. ( x+6)(3x - 2) = x(3x) + x( -2)
Now multiply the inner terms — that is, multiply the second term of the left-hand side binomial by the first term of the right-hand side binomial. In this case, multiply 6 by 3x. (x+ 6)( 3x-2) = x(3x) + x(-2) + 6( 3x)
Next, multiply the last terms of each binomial — that is, multiply the second term of the left-hand side binomial by the second term of the right-hand side binomial. In this case, multiply 6 by -2. (x+ 6)(3x - 2) = x(3x) + x(-2) + 6(3x) + 6( -2)
Commutative Property of Multiplication
a* a=a^2
a(- b)=- a * b
Multiply
Add terms
Diego's parents recently bought a piece of land where they plan to raise pigs. They need to fence off a rectangular pigpen before buying the pigs. A farmer friend told Diego that the dimensions of the pigpen, in yards, vary according to the number of pigs x that are being raised in it.
x= 15
Calculate power and product
Multiply
Add terms
A = 1/2* JK* KL The length of the legs can be written using the given expressions. JK &= x+10 KL &= x^3 + 3x^2 - 18x + 54 Next, substitute the previous expressions into the formula for the area. A(x)=1/2( x+10)( x^3 + 3x^2 - 18x + 54) For simplicity, the product of the two polynomials will be computed first. Then, the resulting polynomial will be multiplied by 12. To perform the polynomial multiplication, the Box Method can be used. To do so, start by drawing a 2* 4 table and writing the row and column labels.
JK* KL= x^4+13x^3+12x^2-126x+540
Distribute 1/2
x= 3
Calculate power and product
Add fractions
Add and subtract terms
Calculate quotient
Add terms
x= -4
Calculate power and product
Calculate quotient
Add and subtract terms
x= -1
Calculate power and product
Add fractions
Subtract terms
Calculate quotient
Add and subtract terms
Izabella bought her nephew a bag of magic grow toys for his birthday. Among the toys, there was a trailer and its rectangular container. Initially, the container was 5 centimeters long, 2 centimeters wide, and 3 centimeters high.
The bag says that when the container is placed underwater, its dimensions will increase according to the following functions. The maximum possible size of the container is reached after being underwater for 9 hours. l(t) &= 14t^2 + 5 [0.2cm] w(t) &= - 19t^2 + 2t + 2 [0.2cm] h(t) &= t + 3 Here, t represents the number of hours the toy has been underwater.
V = w * l * h Therefore, to find a polynomial representing the container's volume after passing t hours underwater, multiply the expressions that model the dimensions of the container. w(t) * l(t) * h(t) ⇓ (1/4t^2 + 5) (-1/9t^2 + 2t + 2) (t + 3) By applying the Commutative Property of Multiplication, the previous product can be rewritten to have the two binomials next to each other. (-1/9t^2 + 2t + 2) (1/4t^2 + 5) (t + 3) Then, these two binomials can be multiplied using the FOIL method. For simplicity, the trinomial will not be written during this multiplication.
Distribute - 19t^2 + 2t + 2
Distribute 14t^3, 34t^2, 5t, & 15
Commutative Property of Multiplication
a^m*a^n=a^(m+n)
Multiply fractions
Simplify quotient and product
Commutative Property of Addition
Add and subtract terms
t= 6
Calculate power
a/c* b = a* b/c
Multiply
Calculate quotient
Add and subtract terms
In all the examples seen throughout this lesson, the product of two or more polynomials has resulted in a new polynomial. This is not a coincidence. In fact, the following property guarantees that multiplying polynomials always produces a polynomial.
Given two polynomials P(x) and Q(x), the product P(x)* Q(x) is always a polynomial.
Multiplying two polynomials produces a new polynomial.
In other words, the polynomials are closed under multiplication.
Given two polynomials P(x) and Q(x), compute their product and find the required information.
Multiply ( x+ 2) by ( x -1)
a^m*a^n=a^(m+n)
a(- b)=- a * b
a * 1=a
Add terms
Distribute 4
Substitute expressions
a^m*a^n=a^(m+n)
Multiply
Commutative Property of Addition
Add and subtract terms
Let's find the required product using the Distributive Property.
Next, we continue simplifying the product by applying the Product of Powers Property and combining like terms.
One of the given polynomials has more than 3 terms. In this case, it could be convenient to use the Box Method to multiply them. Let's start by counting the number of terms.
ccc
Polynomial & & Number of Terms
P(x) & & 3
Q(x) & & 4
We will need to draw a table with three rows and four columns. We will place the terms of P(x) to the left of the table and the term of Q(x) above the table.
Next, we fill in the table's cells by multiplying the terms written on the corresponding borders of the table. For example, the top-left cell corresponds to the product of x^2 and 2x^3. The remaining cells can be filled by following the same procedure. Here, we will apply the Product of Powers Property.
Before, adding all the terms inside the table, let's highlight the like terms.
Finally, to simplify the computations, we will group all the like terms and, after combing them, we will get the result of the polynomial multiplication.
The given polynomials have more than 3 terms. In this case, it could be convenient to use the Box Method to multiply them. Then, let's start by counting the number of terms. ccc Polynomial & & Number of Terms P(x) & & 4 Q(x) & & 4 We will need to draw a table with four rows and four columns. We will place the terms of P(x) to the left of the table and the terms of Q(x) above the table.
Next, we fill in the table's cells by multiplying the terms written on the corresponding borders of the table. For example, the top-left cell corresponds to the product of 2x^4 and x^4. The remaining cells can be filled by following the same procedure. Here, we will apply the Product of Powers Property.
Before, adding all the terms inside the table, let's highlight the like terms.
Finally, to simplify the computations, we will group all the like terms and, after combing them, we will get the result of the polynomial multiplication.
To make the computations easier, let's first find the product of P(x) and Q(x) and then, we will multiply the resulting polynomial by R(x). P(x)Q(x) = (x + 3)(3x^2 - 2x + 4) These two polynomials can be multiplied by using the Distributive Property.
As we can see, the resulting polynomial has 4 terms and, since R(x) has 3 terms, it could be convenient to multiply them using the Box Method. Let's make a 4 by 3 table and write the terms of P(x)Q(x) to the left of the table and the terms of R(x) above the table.
Next, we fill in the table's cells by multiplying the terms written on the corresponding borders of the table. For example, the top-left cell corresponds to the product of 3x^3 and x^2. The remaining cells can be filled by following the same procedure. Here, we will apply the Product of Powers Property.
Before, adding all the terms inside the table, let's highlight the like terms.
Finally, to simplify the computations, we will group all the like terms and, after combing them, we will get the result of the polynomial multiplication.