Let n = 1
then f(1) = 1^1 - 1 + 2 = 2 so it is true for n = 1
for the next number after n ( n+1) we have f(n+1) =
(n+1)^2 - (n+1) + 2

= n^2 + 2n + 1 - n - 1 + 2

= n^2 + n + 2

= n(n+1) + 2

Now n(n+1) must be divisible by 2 because either n is odd and n+1 is even OR n is even and n+1 is odd and odd & even always = an even number.

So the function is divisible by 2 for n+1 We have shown that its true for n = 1 Therefore it must be true for n = 1,2,3,4 ...
True for all positive integers

3 0

3 years ago

(4x-9)(2x^2+2x+1) (4x−9)(2x 2 +2x+1)

tia_tia [17]

Answer:

$(4x-9)^2*(2x^2+2x+1)^2$

Step-by-step explanation:

(((4x-9)•(((2•(x2))+2x)+1))•(4x-9))•((2x2+2x)+1)

] (((4x-9)•((2x2+2x)+1))•(4x-9))•(2x2+2x+1)

Trying to factor by splitting the middle term

Factoring 2x2+2x+1

The first term is, 2x2 its coefficient is 2 .

The middle term is, +2x its coefficient is 2 .

The last term, "the constant", is +1

Step-1 : Multiply the coefficient of the first term by the constant 2 • 1 = 2

Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 2 .

-2 + -1 = -3

-1 + -2 = -3

1 + 2 = 3

2 + 1 = 3

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

((4x-9)•(2x2+2x+1)•(4x-9))•(2x2+2x+1)

Multiplying Exponential Expressions:

4.1 Multiply (4x-9) by (4x-9)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is (4x-9) and the exponents are :

1 , as (4x-9) is the same number as (4x-9)1

and 1 , as (4x-9) is the same number as (4x-9)1

The product is therefore, (4x-9)(1+1) = (4x-9)2

Trying to factor by splitting the middle term

5.1 Factoring 2x2+2x+1

The first term is, 2x2 its coefficient is 2 .

The middle term is, +2x its coefficient is 2 .

The last term, "the constant", is +1

Step-1 : Multiply the coefficient of the first term by the constant 2 • 1 = 2

Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 2 .

-2 + -1 = -3

-1 + -2 = -3

1 + 2 = 3

2 + 1 = 3

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Multiplying Exponential Expressions:

5.2 Multiply (2x2+2x+1) by (2x2+2x+1)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is (2x2+2x+1) and the exponents are :

1 , as (2x2+2x+1) is the same number as (2x2+2x+1)1

and 1 , as (2x2+2x+1) is the same number as (2x2+2x+1)1

The product is therefore, (2x2+2x+1)(1+1) = (2x2+2x+1)2

Final result :

(4x - 9)2 • (2x2 + 2x + 1)2

6 0

2 years ago