- Examples of addition of polynomials
- Add two or more polynomials with more than one variable
- Polynomial addition exercises
- - Exercise 1
- Solution
- - Exercise 2
- Solution to
- References
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.
Figure 1. To add two polynomials it is necessary to order them and then reduce the like terms. Source: Pixabay + Wikimedia Commons.
The terms can be products of real numbers and one or more variables, represented by letters, for example: 3x 2 and -√5.a 2 bc 3 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.
-Terms of equal order are: 5x 3, √2 x 3 and -1 / 2x 3
-Terms of different orders: -2x -2, 2xy -1 and √6x 2 and
It is important to bear 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 polynomials to add, all in the same way, either increasing or decreasing manner, ie with potencies from lowest to highest or vice versa.
- Complete, in case any power is missing in the sequence.
- Reduce like terms.
- Indicate the resulting sum.
Examples of addition of polynomials
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) = 2x 2 - 5x 4 + 2x –x 5 - 3x 3 +12
Q (x) = x 5 - 25 x + x 2
Following the steps described, you begin by ordering them in descending order, which is the most usual way:
P (x) = –x 5 - 5x 4 - 3x 3 + 2x 2 + 2x +12
Q (x) = x 5 + x 2 - 25x
The polynomial Q (x) is not complete, it is seen that there are missing powers with exponents 4, 3 and 0. The latter is simply the independent term, the one without a letter.
Q (x) = x 5 + 0x 4 + 0x 3 + x 2 - 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, like this:
- x 5 - 5x 4 - 3x 3 + 2x 2 + 2x +12
+ x 5 + 0x 4 + 0x 3 + x 2 - 25x + 0 +
--------------------
0x 5 –5x 4 - 3x 3 + 3x 2 - 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, in this way 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.
Add two or more polynomials with more than one variable
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) = 5x 2 - 4y 2 + 8xy - 6y 3
AND:
T (x, y) = ½ x 2 - 6y 2 - 11xy + x 3 and
One of the variables is chosen, for example x to order:
R (x, y) = 5x 2 + 8xy - 6y 3 - 4y 2
T (x, y) = + x 3 y + ½ x 2 - 11xy - 6y 2
Immediately the missing terms are completed, according to which each polynomial has:
R (x, y) = 0x 3 y + 5x 2 + 8xy - 6y 3 - 4y 2
T (x, y) = + x 3 y + ½ x 2 - 11xy + 0y 3 - 6y 2
And you're both ready to reduce like terms:
0x 3 y + 5x 2 + 8xy - 6y 3 - 4y 2
+ x 3 y + ½ x 2 - 11xy + 0y 3 - 6y 2 +
---------------------–
+ x 3 y + 11 / 2x 2 - 3xy - 6y 3 - 10y 2 = R (x, y) + T (x, y)
Polynomial addition exercises
- Exercise 1
In the following sum of polynomials, indicate the term that must go in the blank space to obtain the polynomial sum:
-5x 4 + 0x 3 + 2x 2 + 1
x 5 + 2x 4 - 21x 2 + 8x - 3
2x 5 + 9x 3 -14x
----------------
-6x 5 + 10x 4 -0x 3 + 5x 2 - 11x + 21
Solution
To obtain -6x 5 a term of the form ax 5 is required, such that:
a + 1+ 2 = -6
Thus:
a = -6-1-2 = -9
And the search term is:
-9x 5
-We 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: 13x 4.
-For the powers of x 3 it is immediate that the term must be -9x 3, in this way the coefficient of the cubic term is 0.
-As for the squared powers: a + 8 - 14 = -11 → a = -11 - 8 + 14 = -5 and the term is -5x 2.
-The linear term is obtained by means of 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.
- Exercise 2
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:
Figure 2. A flat terrain is fenced with the shape and dimensions indicated. Source: F. Zapata.
Solution to
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 + length of semicircle + z + length of diagonal + 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
So:
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 = 1/2
These expressions are substituted in that of the perimeter, to obtain:
Perimeter = y + x + πx / 2 + z + 1/2 + z + x + z
Like terms are reduced, since the addition requires that the result be simplified as much as possible:
Perimeter = y + + z + z + z + 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) 2
- Triangle: base x height / 2
Rectangle area
(x + y). (x + z) = x 2 + xz + yx + yz
Semicircle area
½ π (x / 2) 2 = π x 2 /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 = x 2 + xz + yz + x + (π x 2 /8) + zx + ½ ½ zy
And finally all the terms that are similar are reduced:
Total area = (1 + π / 8) x 2 + 3/2 xy + 3 / 2yz + yx
References
- Baldor, A. 1991. Algebra. Editorial Cultural Venezolana SA
- Jiménez, R. 2008. Algebra. Prentice Hall.
- Math is Fun. Adding and subtracting polynomials. Recovered from: mathsisfun.com.
- Monterey Institute. Adding and subtracting polynomials. Recovered from: montereyinstitute.org.
- UC Berkeley. Algebra of polynomials. Recovered from: math.berkeley.edu.