Question

Divide f(x) by d(x), and write a summary statement in the form indicated.

f(x) = x^4 + 4x^3 + 6x^2 + 4x + 5: d(x) = x^2+1


•f(x)=(x^2+1)(x^2+4x+5)+12x-15

•f(x)=(x^2+1)(x^2+4x+5)

•f(x)=(x^2+1)(x^2-4x+5)

•f(x)=(x^2+1)(x^2-4x+5)+12x-15

Answer 1

The polynomial f(x) is divided by d(x) using polynomial division resulting in a quotient of x^2 + 4x + 5 and a remainder of 12x - 15. Thus, the correct summary statement is f(x) = (x^2+1)(x^2+4x+5) + 12x - 15.

Step-by-step explanation:

When dividing the polynomial f(x) = x^4 + 4x^3 + 6x^2 + 4x + 5 by d(x) = x^2 + 1, we perform polynomial long division or synthetic division. Start by determining how many times d(x) goes into the leading term of f(x), which is x^4. This gives us an initial quotient of x^2. Multiplying x^2 by d(x) and subtracting the result from f(x) gives us a new polynomial. We then repeat the process for this new polynomial until we reach a degree less than the degree of d(x). The quotient from the division process is x^2 + 4x + 5 with a remainder of 12x - 15. Therefore, the correct representation of f(x) divided by d(x) is f(x) = (x^2+1)(x^2+4x+5) + 12x - 15.

Answer 2

The correct option is B)
`f(x) =(x^2+1)(x^2+4x+5)`

Step-by-step explanation:

Consider the provided function.

`f(x) = x^4 + 4x^3 + 6x^2 + 4x + 5` and
`d(x) = x^2+1`

We need to divide f(x) by d(x)

As we know: Dividend = Divisor × Quotient + Remainder

In the above function f(x) is dividend and divisor is d(x)

Divide the leading term of the dividend by the leading term of the divisor:
`(x^4)/(x^2)=x^2`

Write the calculated result in upper part of the table.

Multiply it by the divisor:
`x^2(x^2+1)=x^4+x^2`

Now Subtract the dividend from the obtained result:

`(x^4 + 4x^3 + 6x^2 + 4x + 5)-(x^4-x^2)=4x^3+5x^2+4x+5`

Again divide the leading term of the obtained remainder by the leading term of the divisor:
`(4x^3)/(x^2)=4x`

Write the calculated result in upper part of the table.

Multiply it by the divisor:
`4x(x^2+1)=4x^3+4x`

Subtract the dividend:

`(4x^3+5x^2+4x+5)-(4x^3+4x)=5x^2+5`

Divide the leading term of the obtained remainder by the leading term of the divisor:
`(5x^2)/(x^2)=5`

Multiply it by the divisor:
`5(x^2+1)=5x^2+5`

Subtract the dividend:

`(5x^2+5)-(5x^2+5)=0`

Therefore,

Dividend =
`x^4 + 4x^3 + 6x^2 + 4x + 5`

Divisor =
`x^2+1`

Quotient =
`x^2+4x+5`

Remainder = 0

Dividend = Divisor × Quotient + Remainder

`f(x) = (x^2+1)(x^2+4x+5)`

Hence, the correct option is B)
`f(x) =(x^2+1)(x^2+4x+5)`