Questions: Use synthetic division to find the quotient and the remainder. (x^4-4) ÷ (x-3) A. Q(x)=x^3+7x^2+6x+3 ; R(x)=77 B. Q(x)=x^3+3x^2+9x+27 ; R(x)=253 C. Q(x)=x^3+3x^2+9x+27 ; R(x)=77 D. Q(x)=x^3+4x^2+16x+64 ; R(x)=253

Solution Steps

To solve the problem using synthetic division, we need to divide the polynomial \(x^4 - 4\) by \(x - 3\). In synthetic division, we use the root of the divisor, which is 3 in this case, and perform the division process to find the coefficients of the quotient polynomial and the remainder.

We are tasked with dividing the polynomial \( P(x) = x^4 - 4 \) by \( D(x) = x - 3 \). Using synthetic division with the root \( 3 \), we set up the coefficients of \( P(x) \) as \( [1, 0, 0, 0, -4] \).

Through the synthetic division process, we find the coefficients of the quotient polynomial \( Q(x) \). The resulting coefficients are \( [1, 3, 9, 27] \), which correspond to the polynomial: \[ Q(x) = x^3 + 3x^2 + 9x + 27 \]

The remainder from the synthetic division is calculated to be \( 77 \). Thus, we have: \[ R(x) = 77 \]

Final Answer