Question

Преобразуйте в многочлен (v+2)^3-v(v+4)

Answer 1

$(v+2)^3-v(v+4)$
$(v+2)^3=v^3+2v^2+4v^2+8v+4v+8$
$v(v+4)=v^2+4v$
$v^3+6v^2+12v+8-(v^2+4v)=v^3+5v^2+8v+8$

В итоге получаем: $v^3+5v^2+8v+8$

Answer 2

Converting the Expression to a Polynomial

To convert the expression (v + 2)^3 - v(v + 4) into a polynomial, we can follow these steps:

Step 1: Expand the binomial (v + 2)^3 using the binomial theorem. Step 2: Distribute the -v to both terms inside the parentheses (v + 4). Step 3: Simplify the resulting expression.

Let's go through each step in detail.

Step 1: Expand the Binomial (v + 2)^3

To expand the binomial (v + 2)^3, we can use the binomial theorem, which states that for any real numbers a and b and a positive integer n, the expansion of (a + b)^n can be written as:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

where C(n, k) represents the binomial coefficient, which is the number of ways to choose k items from a set of n items.

In our case, a = v and b = 2, and we want to expand (v + 2)^3.

Using the binomial theorem, we can expand (v + 2)^3 as follows:

(v + 2)^3 = C(3, 0) * v^3 * 2^0 + C(3, 1) * v^2 * 2^1 + C(3, 2) * v^1 * 2^2 + C(3, 3) * v^0 * 2^3

Simplifying each term, we get:

(v + 2)^3 = 1 * v^3 * 1 + 3 * v^2 * 2 + 3 * v^1 * 4 + 1 * v^0 * 8

Which further simplifies to:

(v + 2)^3 = v^3 + 6v^2 + 12v + 8

Step 2: Distribute -v to (v + 4)

Next, we distribute -v to both terms inside the parentheses (v + 4):

-v(v + 4) = -v * v - v * 4

Simplifying each term, we get:

-v(v + 4) = -v^2 - 4v

Step 3: Simplify the Resulting Expression

Now, we can simplify the expression by combining like terms:

(v + 2)^3 - v(v + 4) = v^3 + 6v^2 + 12v + 8 - v^2 - 4v

Combining like terms, we get:

(v + 2)^3 - v(v + 4) = v^3 - v^2 + 6v^2 + 12v - 4v + 8

Simplifying further, we have:

(v + 2)^3 - v(v + 4) = v^3 + 5v^2 + 8v + 8

Therefore, the expression (v + 2)^3 - v(v + 4) can be simplified to the polynomial v^3 + 5v^2 + 8v + 8.