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MATH - Coggle Diagram

- - - - Polynomials
        
        They are two or more algebraic expressions (with different literal part) that are being added or subtracted
      - Monomials
        
        It is a single algebraic expression
  - - - In the previous example we have seen the exponents of the polynomial are all consecutive numbers between 0 and 5, but they are in complete disorder
        
        5a2 + 3a3 -a5 + a8 we see that the exponents go up when we see this it is an ordered polynomial
    - - Remember that a polynomial is made up of two or more terms that are being added or subtracted.
        
        3a2 b + 5ab2 -3abc
    - - is complete with respect to a letter when it contains all consecutive exponents of a letter, from highest to lowest
        
        6x3 -5x + 3x5 + x2 -x4 +5 as we see the exponents between 5 and 0 the numbers are consecutive we affirm that it is a complete polynomial
  - - - Geometry, including Trigonometry and Conic Sections
      - Mathematical analysis, which uses letters and
        symbols, and that includes algebra, geometry and calculus
      - Arithmetic
- - - - 5x4 y3 -x4 y3
        
        now we will only subtract the numerical part, that is, 5 - 1, which means that two 4 5x4 y3 - 1x4 y3 = 4x4 y3
        
        the monomial answer will be 4x4 y3
      - 3m2 n + 6m2 n
        
        the monomial answer will be 9m2 n
        
        but we will only add the numerical part, that is, 3 + 6, which gives us 9 3m2 n + 6m2 n = 9m2 n
  - - - To divide polynomials it is not necessary that they be like terms
        
        81a2 b3 c4 d5 between monomial 3b2 c2
        
        81a2 b3 c4 d5 ÷ 3b2 c2
        
        So the answer will be: 27a2 bc2 d5
        
        for letter b we subtract 3 - 2 = 1
        for letter c we subtract 4 - 2 = 2
        
        First we will divide the numerical part as traditionally we do, that is: 81 ÷ 3 = 27
    - - To multiply monomials it will not be necessary for them to be like terms. We can multiply any monomial between them.
        
        a) 5x2 y 5 b) 2x3 y2 z
        
        (5x2 y5) (2x3 y2 z) now we add the exponents of the letter y,
        5 + 2 = 7
        
        (5x2 y3) (2x3 y2 z) the letter z is not repeated so only the
        I will place as is
        
        (5x2 y5) (2x3 y2 z) first for the letter x, we add the exponents 2 + 3 = 5
        
        The answer is 10x5 y7 z
  - - - Let's look at the following pair of algebraic expressions: a) 4x2 y3 b) 2x2 y3
        
        When the literal part in two monomials is equal, then we are talking about like terms
        
        We see that in both expressions the literal part is repeated, in both monomials there are x 2, likewise, in both monomials there is y3
  - - - As in empowerment, in the case of filing we must work separately on the numerical part and the literal part. We will obtain the corresponding root for the numerical part; and in the numerical part we will divide the exponent of each letter by the degree of the radic
        
        In the example, √ (〖16x〗 ^ 4 y ^ 6), we are asked to obtain the square root of the monomial 16x4 y6
        
        Now we divide the exponents by 2 (the index of the radical): x4 ÷ 2 y 6 ÷ 2 = x2 y3
        
        Finally, the answer will be: 4x2y3
        
        We will start with the numerical part: √16 = 4
    - - First we will work the numerical part as we have always done, that is, applying the definition of power. Then we will work with the literal part, in which we will multiply the exponent of each letter by the exponent of the given power.
        
        (3x2 y)4
        
        First we will do the numerical part: 34 = 3 x 3 x 3 x 3 = 81
        
        Finally, the answer will be: 81x8 y4
        
        We are asked to raise the monomial 3x2 and to the 4th power
- - - - (5x2 y +3xy2) (3x3 -2x2y +xy2 -4y3
        
        We do the same with the second term of the first polynomial (3xy2) (3x3) = 9x4 y2; (3xy2) (- 2x2y) = -6x3 y3; (3xy2) (xy2) = 3x2 y4; (3xy2) (- 4y3) = - 12xy5
        
        Now we put together all the results of both multiplications 15x5y -10x4 y2 + 5x3 y3 -20x2 y4 + 9x4 y2 -6x3 y3 + 3x2 y4 + 12xy5
        
        Now we multiply the first term of the first polynomial by each one of the terms of the second polynomial (5x2y) (3x3) = 15x5 y; (5x2y) (-2x2y) = -10x4y2; (5x2y) (xy2) = 5x3y3; (5x2y) (-4y3) = -20x2y4
        
        We look for like terms and add or subtract them: 15x5 y -1x4 y2 -1x3 y3 -17x2 y4 -12xy
  - - - Square of a Binomial
        
        The square of the sum of two terms is equal to the square of the first term plus the double product of both terms plus the square of the second term.
        
        (5x+7)2 = (5x)2 + 2(5x) (7) + (7)
        
        The double product of both terms is: 2 (5x) (7) = 70x
        
        The square of the second term is: (7) 2 = 49
        
        The square of the first term is: (5x1) 2 = 25x2
        
        Finally, the answer will be: (5x +7) 2 = 25x2 + 70x + 4
      - The Sum by the Difference of Two Quantities
        
        The sum of two terms multiplied by their difference is equal
        to the difference of the squares of both terms
        
        (4a +7y3) (4a -7y3) = (4a)2 - (7y3)2 = 16a2- 49y6
        
        The square of the second term is: (7y3) 2 = 49y
        
        Finally, the answer will be: (4a + 7y3) (4a -7y3) = 16a2- 49y6
        
        The square of the first term is: (4a1) 2 = 16a2
  - - - To perform a subtraction, the procedure is similar, but we must be very careful with the signs. Let's say that now we are going to subtract: 5x2 y + 3xy2 - (3x3 -2x2 y + xy2 -4y3
        
        The parenthesis of the second polynomial is eliminated, changing the sign of its terms: 5x2y + 3xy2 -3x3 + 2x2 and -xy2 + 4y
        
        Now we just look for the like terms and carry out the corresponding operations: 5x2y + 2x2y + 3xy2 -xy2 -3x3
        
        4y3
        
        The answer is 7x2y + 2xy2 -3x3 + 4y
      - Let's add the polynomial 5x2 y + 3xy2 and the polynomial 3x3 -2x2y + xy2 -4y3
        
        Non-like terms will be included in the final answer as is
        
        We introduce the partial results in our polynomial response: 3x2y + 4xy2 + 3x3 -4y3
        
        We operate the terms with xy2: 3xy2 + 1xy2 = 4xy
        
        We operate the terms with x2y: 5x2 y -2x2y = 3x2y
        
        Now we must see if there are like terms and we add them between them
        
        Then the sum will be: 5x2 y + 3xy2 + 3x3 -2x2 y + xy2 - 4y3
  - - - (2a + 4b)3 = (2a)3 +3(2a)2(4b) +3(2a) (4b)2 + (4b)3
        
        The triple of the first term times the square of the second term:
        3 (2a) (4b1) 2 = 3 (2a) (16b2) = 96ab
        
        The cube of the second term is: (4b1) 3 = 64b
        
        The triple of the square of the first term by the second term:
        3 (2a1) 2 (4b) = 3 (4a2) (4b) = 48a2
        
        Finally, the answer will be: (2a + 4b) 3 = 8a3 + 48a2b +
        96ab2 + 64b3
        
        The cube of the first term is: (2a1) 3 = 8a3
- - - - 2x -[x -(x -50)] = x - (800 -3x
        
        2x -50 = 4x -800 Ahora quitamos los corchetes.
        
        2x -4x = -800 +50 We transpose the terms, using the criterion of inverse operations.
        
        2x - [50] = 4x -800 We reduce like terms
        
        -2x = -750 Again we reduce like terms
        
        2x - [x -x +50] = x -800 + 3x First we remove the parentheses
        
        x = -750 / -2 = 375 We solve for x going to divide to -2, then we simplify
  - - - 5(x -3) -(x -1) = (x +3) -1
        
        5x -x -x = 3 -10 +15 We take the like terms to one side of the equality, and the independent terms to the other side (we use inverse operations.)
        
        3x = 9 We reduce like terms on both sides of the equality
        
        5x -15 -x -1 = x +3 -10 We solve the indicated product, and additionally we eliminate the parentheses
        
        x = 9/3 = 3 We solve for x passing 3 to divide, then we simplify
  - - - Solve the equation 2x - 3 = 53
        
        Now we add and subtract the like and independent terms, leaving the equation like this: 2x = 5
        
        Now we have the number 2 that is multiplying the variable or unknown x. So we transpose it (to number 2) to the other side of the equality. If the “2” was multiplying, it will happen by dividing (the inverse operation). The equation would look like this: X = 56/2
        
        We must have the letters on one side and the numbers on the other side of the equality (=). To do this, we transpose the terms to the corresponding member (side). When transposing a term, it happens with the changed sign. For example: when transposing the -3 to the other side of the equality, it will be +3. We will then have 2x = 53 +3
        
        We perform the division and we will obtain the following: x = 28. This is the answer and it tells us that the value of x is 28. The answer is also an equation since it tells us that the quantity "x" is equal to the quantity "28"