Answer:
The correct option is B) [tex]f(x) =(x^2+1)(x^2+4x+5)[/tex]
Step-by-step explanation:
Consider the provided function.
[tex]f(x) = x^4 + 4x^3 + 6x^2 + 4x + 5[/tex] and [tex]d(x) = x^2+1[/tex]
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:[tex]\frac{x^4}{x^2}=x^2[/tex]
Write the calculated result in upper part of the table.
Multiply it by the divisor: [tex]x^2(x^2+1)=x^4+x^2[/tex]
Now Subtract the dividend from the obtained result:
[tex](x^4 + 4x^3 + 6x^2 + 4x + 5)-(x^4-x^2)=4x^3+5x^2+4x+5[/tex]
Again divide the leading term of the obtained remainder by the leading term of the divisor: [tex]\frac{4x^3}{x^2}=4x[/tex]
Write the calculated result in upper part of the table.
Multiply it by the divisor: [tex]4x(x^2+1)=4x^3+4x[/tex]
Subtract the dividend:
[tex](4x^3+5x^2+4x+5)-(4x^3+4x)=5x^2+5[/tex]
Divide the leading term of the obtained remainder by the leading term of the divisor: [tex]\frac{5x^2}{x^2}=5[/tex]
Multiply it by the divisor: [tex]5(x^2+1)=5x^2+5[/tex]
Subtract the dividend:
[tex](5x^2+5)-(5x^2+5)=0[/tex]
Therefore,
Dividend = [tex]x^4 + 4x^3 + 6x^2 + 4x + 5[/tex]
Divisor = [tex]x^2+1[/tex]
Quotient = [tex]x^2+4x+5[/tex]
Remainder = 0
Dividend = Divisor × Quotient + Remainder
[tex]f(x) = (x^2+1)(x^2+4x+5)[/tex]
Hence, the correct option is B) [tex]f(x) =(x^2+1)(x^2+4x+5)[/tex]
