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Concept Map 3rd learning cycle -Michelle Báez- - Coggle Diagram
Concept Map 3rd learning cycle -Michelle Báez-
Multiplication of polynomials
Definition:Is a process for multiplying together two or more polynomials.
Explanation:Multiply each term in one polynomial by each term in the other polynomial. add those answers together, and simplify if needed.
Example: 3x²(4x² – 5x + 7)
Monomial x monomial: (x²)(x⁷)=x⁹
How to solve it?:We add the exponents
Monomial x polynomial:(3x)(2x+3y+5z)=6x²+9xy+15xz
How to solve it?:We multiply 3x by 2x+3y+5z and ge wet the final result
Polynomial x polynomial:(x²+4x+3)(2x²-9x+7)=2x⁴-x³-23x²+x+21
How to solve it?:We multiply x² by 2x²-9x+7, then we multiply 4x by 2x²-9x+7 and finally 3 by 2x²-9x+7. We substract the like terms until we get the final result.
Special Products
Definition:Special products, are polynomials of two terms (binomials) elevated to the square, or the product of two binomials.
Explanation:Special products are simply special cases of multiplying certain types of binomials together.
Examples: (x+a)(x-a) or (x+a)²
(a+b)²=(a+b)(a+b)
Example:(a+2)²=(a+2)(a+2)=a²+4a+4
How to solve it?:Actually (a+2)² is the same as (a+2)(a+2) but in a shorter version, so we multiply a by a+2 and 2 by a+2 and add the like terms and we got the result
(a-b)²=(a-b)(a- b)
Example:(x-2)²=(x-2)(x-2)=x²-4x+4
How to solve it?:Like I said above (x-2)² is the same as (x-2)(x-2) but in a shorter version and now with negatives,so we multiply x by x-2 and -2 by x-2 and we substract the like terms to got the final result
(a+b)³=(a+b)(a+b)(a+b)
Example:(x+6)³=(x+6)(x+6)(x+6)=x³+18x²+108x+216
How to solve it?(x+6)³ is the same as (x+6)(x+6)(x+6) so we first multiply x by x+6, then 6 by x+6 and add the like terms. Finally that result we multiply by x+6, add the like terms and we get the final answer
(a-b)³=(a-b)(a-b)(a-b)
Example:(x-4)³=(x-4)(x-4)(x-4)=x³-12x²+48x-64
How to solve it?(x-4)³ is the same as (x-4)(x-4)(x-4) so we first multiply x by x-4,then -4 by x-4 and we substract the like terms. Finally that result we multiply by x-4, substract the like terms and we get the final answer
(a+b)(a-b)=(a)(a)+(a)(-b)+(b)(a)+(b)(-b)=a²-b²
Example:(x+5)(x-5)=(x)(x)+(x)(-5)+(5)(x)+(5)(-5)=x²-25
How to solve it?(x+5)(x-5) can be solved as multiplying this way (x)(x)+(x)(-5)+(5)(x)+(5)(-5), then we substract the like terms and we got the final answer
(x+a)(x+b)=(x)(x+b)+(a)(x+b)=x²+xb+ax+ab
Example:(x+2)(x+6)=(x)(x+6)+(2)(x+6)=x²+8x+12
How to solve it?(x+2)(x+6) can be solved as multiplying this way:x by x+6 and 2 by x+6, then we add like terms to obtain the final answer
Factorization
Definition:Consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
Explanation:Is when you break a number down into smaller numbers that, multiplied together, give you that original number. When you split a number into its factors or divisors, that's factorization.
Example:Factorization of the number 12 might look like 3 times 4.
Quadratic polynomials
x²+(a+b)x+ab
Example:x²+7x+12=(x+4)(x+3)
How to solve it?Like we saw in the first operation x²+(a+b)x+ab, so we factorize x, so x by x will give us x², then we find to numbers that added gave us 7 and multiplied 12 and that numbers are 4 and 3 and that's how we obtain the factorization
Perfect square trinomials
(a²+2ab+b²)
Example:(1x²+6x+9)=(x+3)²
How to solve it?To check the factorization we can use the (a²+2ab+b²) formula so if we substitute we got x²+2(3x)+3²,that can be simplified as (x+3)² so the factorization is correct
Difference of squares
(a²-b²)=(a+b)(a-b)
Example:(x²-64)=(x+8)(x-8)
How to solve it?Actually is very easy cause we only need to find a number that multipied by itself gives 64, that will be 8 and then x² will be x multiplied by itself. Now we put the result one positive and the other negative and we got our final answer
Division of polynomials
Definition:Is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique.
Explanation:When dividing a monomial by another monomial, we divide the coefficients and apply the quotient law x^m ÷ x^n = x^m – ^n to the variables.
Example:Division of polynomial by monomial
Monomial ÷ monomial
Example:16m⁶n⁴/4n³=4m⁶n
How to solve it?We divide 16 over 4 and it gaves us 4, then we substract the exponents and we got our final result.
Polynomial ÷ monomial
Example:(x³-4x²+x)/x=x²-4x
How to solve it?: First we divide x³ by x, then -4x² by x and finally x by x. The final result substracting already the exponents will be x²-4x
Polynomial ÷ polynomial
TEACHER I APOLOGIZE BUT I DIDN'T FOUND ANY EASY EXAMPLE AND ALSO I DIDN'T UNDERSTAND HOW TO SOLVE IT.
THANKS FOR UNDERSTANDING
