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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.
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Commutative Property of Addition
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Commutative Property of Addition
Previously, polynomials were added by using the horizontal format. The vertical format of polynomial addition or subtraction will now be presented. Note that for this method, the like terms should be aligned vertically. &-2x^2 + x + 3 +& 2x^2 + 4x - 10 & 0x^2 + 5x - 7 By comparing the results, it can be concluded that no matter which format of addition is used, the sums are the same.
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.
B&=- 0.023n^3+0.08n^2+0.1n+15 R&=- 0.41n^2+1.3n+36 Here, n=1 represents the first year in the 6-year period.
Remove parentheses
Commutative Property of Addition
Add and subtract terms
a+(- b)=a-b
- &0.41n^2+1.3n+36 + (- 0.023n^3+&0.08n^2+0.1n+15) - 0.023n^3-&0.33n^2+1.4n+51
B + R = - 0.023n^3 - 0.33n^2 + 1.4n + 51 R + B = - 0.023n^3 - 0.33n^2 + 1.4n + 51 As can be seen, these sums are the same polynomial. Recall that addition of real numbers is commutative. Since adding polynomials comes down to adding the coefficients of like terms, which are real numbers, it can be concluded that the addition of polynomials is also commutative.
- 0.023n^3-0.33n^2+1.4n+51 Therefore, the degree of this polynomial is 3. The leading coefficient of a polynomial is the coefficient of the term with the highest degree. Since the first term has the highest degree, the leading coefficient is - 0.023. - 0.023n^3-0.33n^2+1.4n+51
Substitute expressions
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Commutative Property of Addition
Add and subtract terms
- 16x^2+ 0x&+100 - - 16x^2-45x &+180 45x &- 80
T(x)-P(x)&=- 45x+80 P(x)-T(x)&=45x-80 The polynomials are different. This suggests that when subtracting polynomials, their order does matter. In other words, the subtraction of polynomials is not commutative.
T(x)-P(x)&=- 45x^1+80 P(x)-T(x)&=45x^1-80 The degree of each polynomial is 1. Next, recall that the leading coefficient of a polynomial is the coefficient of the term with the highest degree. T(x)-P(x)&= - 45x+80 P(x)-T(x)&= 45x-80 In the first polynomial, the leading coefficient is - 45, while it is 45 in the second polynomial.
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
Write the polynomial, in standard form, that represents the perimeter of the quadrilateral shown in the diagram.
The perimeter of a polygon is calculated by adding all the side lengths of the polygon. Therefore, to find the perimeter of the given quadrilateral, we will add all four side lengths together. Then we will simplify the resulting expression by adding and subtracting like terms. Let's do it!
We found that the perimeter of the given quadrilateral is 16x-7 units. Perimeter 16x-7 Keep in mind that although it is not stated, x must be a real number such that all four side lengths and the perimeter are positive. Side Length I:& 3x>0 Side Length II:& - x^2+4x-3>0 Side Length III:& 3x-1>0 Side Length IV:& x^2+6x-3>0 Perimeter:& 16x-7>0 We should disregard all values of x that make any of the above expressions non-positive. For example, x cannot be non-negative because if that were the case, the expression 3x would be non-positive.
Determine whether each statement is true or false.
By its own definition, a binomial is the sum of two monomials. Also, we know that the degree of a binomial is the greatest degree of the monomials. If the degree of a monomial is zero, then it is a constant. Therefore, if the degree of a binomial is zero, then it is the sum of two constants. Binomial = Constant + Constant Since the sum of two constants is also a constant, the resulting expression has only one term and therefore is not a binomial — in other words, it is a monomial. Constant=Monomial As such, we can conclude that a binomial cannot have a degree of zero. This means that the given statement is false.
To verify whether the given statement is true or false, let's consider two arbitrarily chosen polynomials.
P(x) = x^2 + 1 and Q(x) = x^2 - 1
First, let's calculate P(x)-Q(x).
Next, let's calculate the subtraction in the opposite order, Q(x)-P(x).
As we can see, the results are different. Therefore, the order in which polynomials are subtracted is relevant. This means that the given statement is false.