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Concept Map of the 3rd Cycle - Coggle Diagram
Concept Map of the 3rd Cycle
Multiplication of Polynomials
How do we multiply polynomials?
First multiply the coefficients and then multiply the variables by adding the exponents.
(3a) (-2a) = -6a^2
Note that when you multiply monomials with same base, you can add their exponents.
This is called the Product of Powers Property.
x^4
x^3
= x^7
Examples:
Monomial * Monomial:
5x * 7y = 35xy
Multiply the monomial by the other monomial including exponents, the exponents they add them together.
Monomial * Polynomial:
5x(x^2+7x+12) = (5x) (x^2) + (5x) (7x) + (5x) (12) = 5x^3 + 35x^2 +60x
Distribute the monomial to every term of the polynomial. In this case, we need to distribute the 5x.
Combine like terms. In these case are not like terms.
Polynomial * Polynomial:
(x^2 +7x +12) (x^2 -9x +20) =
Distribute each term of the first polynomial to every term of the second polynomial removing the parentheses. In this case, we need to distribute the x^2, 7x and 12.
x^2 (x^2 -9x +20) +7x (x^2 -9x +20) +12 (x^2 -9x +20)
Combine like terms.
x^4 -9x^3 + 20x^2 +7x^3 -63x^2 +140x +12x^2 -108x +240
x^4 -2x^3 -31x^2 +32x +240
Like Terms:
-9x^3 +7x^3 = -2x^3
20x^2 -63x^2 +12x^2 = -31x^2
140x -108x = 32x
Special Products
Special products are simply special cases of multiplying certain types of binomials together.
Examples: (a+b)^2
The first is “a” (the first term of the polynomial) and the second is “b” (the second term of the polynomial).
Convert the polynomial square, into Polynomial * Polynomial.
Solve it as (Binomial * Binomial) and combine like therms.
(a+b)^2 = (a+b)(a+b) = a^2 + ab +ab +b^2 = a^2+2ab+b^2
(a-b)^2 = (a-b) (a-b) = a^2 - ab - ab + b^2 = a^2-2ab-b^2
Examples: (a+b)^3
The same as square but with cube and converting it into 3 binomials.
(a+b)^3 = (a+b)(a+b)(a+b)= a^2 + ab +ab +b^2 = (a^2+2ab+b^2) (a+b) = (a+b) (a^2) + (a+b) (2ab) + (a+b) (b^2)
a+b * a^2 = a^3 + a^2b
a+b * 2ab = 2a^2 b + 2ab^2
a+b * b^2= ab^2 + b^3
a^3 + a^2 b + 2a^2 b +2ab^2 + ab^2 + b^3
a^3 + 3a^2 b + 3ab^2 + b^3
Like Terms:
a^2 b + 2a^2 b = 3a^2 b
2ab^2 + ab^2 = 3ab^2
Firstly solve the first 2 binomial and after that distribute the 3rd binomial with the Polynomial and combine like terms.
Examples: (x+a) (x+b)
Distribute each term of the first binomial to every term of the second binomial removing the parentheses. In this case, we need to distribute the x and a.
(x+a) (x+b) = x^2 + bx + ax +ab
And with: (a+b)(a-b)
= a^2 - ab +ab - b^2
It removes the middle of the result because its 0 the result of the like terms.
= a^2 - b^2
Factorization
Is a technique that consists of descompose an algebraic expression in product form.
Quadratic Polynomials [x^2 + (a+b) x + ab]
12x^2 -13x -4
(3x) (4x +1) - (4)(4x +1)
12x^2 + 3x = (3x) (4x +1)
=12x^2 + 3x -16x -4
16x -4 = 4(4x +1)
(4x+1) (3x-4)
Is a polynomial function with one or more variables in which the highest degree term is of the second degree or more.
Perfect square trinomials (a^2 + ab + b^2)
(3x + 2y)^2
(3x + 2y)(3x + 2y)=
9x^2 + 6xy + 6xy + 4y^2
9x^2 + 12xy + 4y^2
An expression obtained from the square of binomial equation is a perfect square trinomial.
Differences of squares (a^2 - b^2)
9x^2 -16
(3x)^2 - 4^2
(3x+4) (3x-4)
Determine what numbers squares will produce the desired results.
Division of Polynomials
How do we divide Polynomials?
First divide the denominator with each numerator including exponents, but exponents we have to substract them.
Examples:
(x^3 -4x^2 +x) / (x) = x^2 -4x
x^3 / x = x^2
-4x^2 / x = -4x
x / x = 0
(6a^3 -8a^2 b +20 ab^2) / (-2a) = -3a^2+ 4ab -10b^2
-8a^2 b / -2a = 4ab
20ab^2 / -2a = -10b^2
6a^3 / -2a = -3a^2
Monomial / Monomial
16m^6 n^4 / 4n^3
4m^6 n
Divide the monomial by the other monomial including exponents, the exponents they substract them together.
Monomial / Polynomial
You can split the problem into pieces by putting each term in the numerator over the denominator.
(6a^8 b^8 -3a^6 b^6 -a^2 b^3) / (3a^2 b^3) = 2a^6 b^5 -a^4 b^3 - 3
6a^8 b^8 / 3a^2 b^3 = 2a^6 b^5
-3a^6 b^6 / 3a^2 b^3 = -a^4 b^3
-a^2 b^3 / 3a^2 b^3 = -3
Polynomial / Polynomial
Make sure the polynomial is written in descending order. Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol.
Multiply the answer obtained by the polynomial in front of the division symbol.
Substract and bring down th next term.
Repeat the steps until there are no more terms to bring down.
Write the final answer.
Mateo Q. 08/12/20 Final Proyect - Math 1
