Question

From a polynomial that represents the product of three consecutive odd integers if the first one is
2x + 1​

Answer 1

Answer:

(8x³ + 36x² + 46x + 15) is the polynomial that represents the product of 3 consecutive odd integers, when the first integer is (2x + 1)

Step-by-step explanation:

We have,

The first integer to be (2x + 1)

Now,

We must notice that,

Odd number + 2 = Next odd number

For ex:-

5 + 2 = 7

Here, the next odd number after 5 is 7.

So, when we add 2 to an odd number it will always give us the next odd number.

Hence,

Next odd number after (2x + 1) is,

(2x + 1) + 2

= 2x + 1 + 2

= (2x + 3)

Then,

The next odd number after (2x + 3) is,

(2x + 3) + 2

= 2x + 3 + 2

= (2x + 5)

Hence,

Our 3 consecutive odd integers are (2x + 1), (2x + 3), and (2x + 5)

Now,

We need to find the product of the 3 consecutive odd integers.

Thus,

(2x + 1)(2x + 3)(2x + 5)

Since, Multiplication is Associative, we can start from any integer,

= [(2x + 1)(2x + 3)] × (2x + 5)

Using Distributive Property,

= [(2x × 2x) + (2x × 3) + (1 × 2x) + (1 × 3)] × (2x + 5)

= [4x² + 6x + 2x + 3] × (2x + 5)

= (4x² + 8x + 3)(2x + 5)

Again, Using Distributive Property,

= [(4x² × 2x) + (4x² × 5) + (8x × 2x) + (8x × 5) + (3 × 2x) + (3 × 5)]

= 8x³ + 20x² + 16x² + 40x + 6x + 15

= 8x³ + 36x² + 46x + 15

Hence,

8x³ + 36x² + 46x + 15 is the polynomial that represents the product of 3 consecutive odd integers, when the first integer is (2x + 1).

Hope it helped and believing you understood it........All the best