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| 9 Theory slides |
| 9 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.
Remove parentheses
Commutative Property of Addition
Remove parentheses
Distribute -1
Commutative Property of Addition
Remove parentheses
Commutative Property of Addition
Add and subtract terms
Dylan's family is moving into an apartment. He is curious about how many people live in apartements as opposed to houses. He finds a study that states that during a 6-year period, the amounts of money in millions of dollars spent on buying houses B and renting houses R by United States residents are modeled by the following two polynomials.
Remove parentheses
Commutative Property of Addition
Add and subtract terms
a+(-b)=a−b
Substitute expressions
Remove parentheses
Distribute -1
Commutative Property of Addition
Add and subtract terms
Consider the sum or difference of the given polynomials. Identify their degree or leading coefficient.
Smart Applications of Math.
In order to add or subtract polynomials, align the like terms vertically and then perform the addition or subtraction.
First, rewrite Q(x) in standard form. Then add and subtract the polynomials by adding and subtracting their like terms.
Substitute expressions
Commutative Property of Addition
Add and subtract terms
Substitute expressions
Remove parentheses
Distribute -1
Commutative Property of Addition
Add and subtract terms
We want to find the leading coefficient of the sum of P(x)+Q(x). Let's start by calculating the sum of the polynomials.
We found an expression for the sum P(x)+Q(x). The leading coefficient is the number that multiplies the power with the greatest exponent. P(x)+Q(x)= 4x+2 The leading coefficient of the sum is 4.
In Part A we calculated the sum of the polynomials. Now we want to find the degree of the expression. Recall that the degree of a polynomial is the exponent of the greatest power of the variable, which in this case is x. When a variable has no exponent, it means that the exponent is 1.
P(x)+Q(x)=4x+2 ⇕ P(x)+Q(x)=4x^1+2
The degree of the expression that represents the sum is 1.
We want to find the leading coefficient of the difference of P(x)-Q(x). We will start by calculating the difference of the polynomials. Remember to write both polynomials in parentheses. Then we can distribute - 1 and simplify the expression.
We found an expression for the difference P(x)-Q(x). The leading coefficient is the number that multiplies the power with the greatest exponent. P(x)-Q(x)= - 6x^2-4 The leading coefficient of the difference is - 6.
In Part A we calculated the difference of the polynomials. Now we want to find the degree of the expression. Recall that the degree of a polynomial is the exponent of the greatest power of the variable, which in this case is x.
P(x)-Q(x)=- 6x^2-4
The degree of the difference is 2.
In the following cases, write an expression in standard form for P(x)+Q(x).
To add the given polynomials, we will start by writing both of them in parentheses. Then we will remove the parentheses and rearrange the terms by using the Commutative Property of Addition. Finally, we will add and subtract like terms. Let's do it!
The sum of the polynomials is -2x+7.
As in Part A, we will start by writing both polynomials in parentheses. Then we will remove the parentheses and rearrange the terms by using the Commutative Property of Addition. Finally, we will add and subtract like terms.
The sum of the polynomials is 3x^3-x^2+3x-9.
In the following cases, write an expression in standard form for P(x)−Q(x).
To subtract the given polynomials, we will start by writing both of them in parentheses. Then we will distribute - 1 to Q(x), remove the parentheses, and rearrange the terms by using the Commutative Property of Addition. Finally, we will add and subtract like terms. Let's do it!
The difference of the polynomials is 10x+3.
As in Part A, we will write both polynomials in parentheses. Then we will distribute - 1 to Q(x), remove the parentheses, and rearrange the terms by using the Commutative Property of Addition. Finally, we will add and subtract like terms.
The difference of the polynomials is 3x^3-7x^2-x+9.
We want to write a polynomial that represents how much more it costs to make x sweaters than x shirts. To do so, we will subtract the cost to make x shirts from the cost to make x sweaters. Let's write an expression to illustrate this situation. We will start by isolating each expression in parentheses first. x sweaters -& xshirts [0.5em] (6x+8) -& (5+4x) We can simplify this expression by distributing - 1 to the expression representing the cost to make the shirts in order to remove the parentheses, then using the Commutative Property of Addition to rearrange the terms. Finally, we will add and subtract like terms to simplify the resulting polynomial. Let's do it!
We can conclude that the difference between the cost of making x sweaters and the cost of making x shirts is 2x+3 dollars.
Complete the statement with never, sometimes, or always.
Every term of a polynomial is blankspace a monomial. |
Let's start by recalling the definitions of a monomial and a polynomial.
Monomial | Polynomial | |
---|---|---|
Definition | A monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents. | A polynomial is a monomial or a sum of monomials. Each monomial is called a term of the polynomial. |
Degree | The degree of a monomial is the sum of the exponents of the variables in the monomial. The degree of a non-zero constant term is 0. The constant 0 does not have a degree. | The degree of a polynomial is the greatest degree of its terms. |
Standard Form | - | A polynomial in one variable is in standard form when the exponents of the terms decrease from left to right. |
Leading Coefficient | The coefficient of the monomial. | When a polynomial is written in standard form, the coefficient of the first terms is the leading coefficient. |
Since a polynomial is a monomial or a sum of monomials, we can conclude that every term of a polynomial is always a monomial.
Every term of a polynomial is always a monomial.