Questions: Use synthetic division to find the quotient and the remainder. (x^5+x^3-x) ÷ (x-4) Q(x)= R(x)=

Transcript text: Use synthetic division to find the quotient and the remainder. \[ \begin{array}{l} \left(x^{5}+x^{3}-x\right) \div(x-4) \\ Q(x)=\square \end{array} \] $\square$ \[ R(x)= \] $\square$

Solution

Solution Steps

Step 1: Write down the coefficients of the polynomial in descending order.

The coefficients are: [1, 0, 1, 0, -1, 0].

Step 2: Place the constant $k = 4$ outside the synthetic division symbol.

Step 3: Perform the synthetic division.

The synthetic division process is carried out as follows: Coefficient of $x^4$: 1 Coefficient of $x^3$: 4 Coefficient of $x^2$: 17 Coefficient of $x^1$: 68 Coefficient of $x^0$: 271 Remainder: 1084

Step 4: Write down the quotient and remainder.

The quotient polynomial is: $Q(x) = x^4 + 4x^3 + 17x^2 + 68x^1 + 271x^0$. The remainder is: 1084.