The sum of polynomials is the operation that consists of adding two or more polynomials, resulting in another polynomial. To carry it out, it is necessary to add the terms of the same order of each of the polynomials and indicate the resulting sum.
Let's first briefly review the meaning of "terms of the same order." Any polynomial is made up of additions and / or subtractions of terms.
The terms can be products of real numbers and one or more variables, represented by letters, for example: 3xtwo and -√5.atwobc3 are terms.
Well, the terms of the same order are those that have the same exponent or power, although they may have a different coefficient.
-Equal order terms are: 5x3, √2 x3 and -1 / 2x3
-Different order terms: -2x-two, 2xy-1 and √6xtwoY
It is important to keep in mind that only terms of the same order can be added or subtracted, an operation known as reduction. Otherwise the sum is simply left indicated.
Once the concept of terms of the same order has been clarified, the polynomials are added following these steps:
-Order first the polynomials to be added, all in the same way, either in an increasing or decreasing way, that is, with the powers from lower to higher or vice versa.
-To complete, in case any power is missing in the sequence.
-Reduce like terms.
-Indicate the resulting sum.
Article index
We will start by adding two polynomials with a single variable called x, for example the polynomials P (x) and Q (x) given by:
P (x) = 2xtwo - 5x4 + 2x -x5 - 3x3 +12
Q (x) = x5- 25 x + xtwo
Following the steps described, you start by ordering them in descending order, which is the most usual way:
P (x) = -x5- 5x4 - 3x3 + 2xtwo + 2x +12
Q (x) = x5+ xtwo - 25x
The polynomial Q (x) is not complete, it is seen that powers with exponents 4, 3 and 0 are missing. The latter is simply the independent term, the one that has no letter.
Q (x) = x5+ 0x4 + 0x3 + xtwo - 25x + 0
Once this step is done, they are ready to add. You can add the like terms and then indicate the sum, or place the ordered polynomials one below the other and reduce by columns, in this way:
- x5 - 5x4 - 3x3 + 2xtwo + 2x +12
+ x5 + 0x4 + 0x3 + xtwo - 25x + 0 +
--
0x5-5x4 - 3x3 +3xtwo - 23x + 12 = P (x) + Q (x)
It is important to note that when it is added, it is done algebraically respecting the rule of signs, thus 2x + (-25 x) = -23x. That is, if the coefficients have a different sign, they are subtracted and the result carries the sign of the greater.
When it comes to polynomials with more than one variable, one of them is chosen to order it. For example, suppose you ask to add:
R (x, y) = 5xtwo - 4ytwo + 8xy - 6y3
Y:
T (x, y) = ½ xtwo- 6ytwo - 11xy + x3Y
One of the variables is chosen, for example x to order:
R (x, y) = 5xtwo + 8xy - 6y3 - 4ytwo
T (x, y) = + x3y + ½ xtwo - 11xy - 6ytwo
The missing terms are immediately completed, according to which each polynomial has:
R (x, y) = 0x3y + 5xtwo + 8xy - 6y3 - 4ytwo
T (x, y) = + x3y + ½ xtwo - 11xy + 0y3 - 6ytwo
And you're both ready to reduce like terms:
0x3and + 5xtwo + 8xy - 6y3 - 4ytwo
+ x3y + ½ xtwo - 11xy + 0y3 - 6ytwo +
-
+ x3and + 11 / 2xtwo - 3xy - 6y3 - 10ytwo = R (x, y) + T (x, y)
In the following sum of polynomials, indicate the term that must go in the blank space to obtain the polynomial sum:
-5x4 + 0x3 + 2xtwo + 1
x5 + 2x4 - 21xtwo + 8x - 3
2x5 +9x3 -14x
-
-6x5+10x4 -0x3 + 5xtwo - 11x + 21
To get -6x5 a term of the form ax is required5, such that:
a + 1+ 2 = -6
Therefore:
a = -6-1-2 = -9
And the search term is:
-9x5
-Proceed in a similar way to find the rest of the terms. Here's the one for exponent 4:
-5 + 2 + a = 10 → a = 10 + 5-2 = 13
The missing term is: 13x4.
-For the powers of x3 it is immediate that the term must be -9x3, thus the coefficient of the cubic term is 0.
-Regarding the squared powers: a + 8 - 14 = -11 → a = -11 - 8 + 14 = -5 and the term is -5xtwo.
-The linear term is obtained by a +8 -14 = -11 → a = -11 + 14 - 8 = -5, the missing term being -5x.
-Finally the independent term is: 1 -3 + a = -21 → a = -19.
A flat terrain is fenced as shown in the figure. Find an expression for:
a) The perimeter and
b) Its area, in terms of the indicated lengths:
The perimeter is defined as the sum of the sides and contours of the figure. Starting in the lower left corner, clockwise, we have:
Perimeter = y + x + semicircle length + z + diagonal length + z + z + x
The semicircle has a diameter equal to x. Since the radius is half the diameter, you have to:
Radius = x / 2.
The formula for the length of a complete circumference is:
L = 2π x Radius
Then:
Length of semicircle = ½. 2π (x / 2) = πx / 2
For its part, the diagonal is calculated with the Pythagorean theorem applied to the sides: (x + y) which is the vertical side and z, which is the horizontal:
Diagonal = [(x + y)two + ztwo]1/2
These expressions are substituted in the one for the perimeter, to obtain:
Perimeter = y + x + πx / 2 + z + [(x + y)two + ztwo]1/2+ z + x + z
Like terms are reduced, since the addition requires that the result be simplified as much as possible:
Perimeter = y + [x + π (x / 2) + x] + z + z + z + [(x + y)two + ztwo]1/2 = y + (2 + π / 2) x + 3z
Solution b
The resulting area is the sum of the area of the rectangle, the semicircle, and the right triangle. The formulas for these areas are:
-Rectangle: base x height
-Semicircle: ½ π (Radius)two
-Triangle: base x height / 2
Rectangle area
(x + y). (x + z) = xtwo + xz + yx + yz
Semicircle area
½ π (x / 2)two = π xtwo / 8
Triangle area
½ z (x + y) = ½ zx + ½ zy
Total area
To find the total area, the expressions found for each partial area are added:
Total area = xtwo + xz + yx + yz + (π xtwo / 8) + ½ zx + ½ zy
And finally all the terms that are similar are reduced:
Total area = (1 + π / 8) xtwo + 3/2 xy + 3 / 2yz + yx
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