Question

use synthetic division to divide the two polynomials.$\begin{array}{r}z-7enclose{longdiv}{4z^{5}-25z^{4}-23z^{3}+11z^{2}+17z+54}end{array}$is the divisor given in (x - r) form? yeshow many terms are in the dividend?enter the quotient and remainder.quotient + remaindersynthetic division table

Explanation:

Step1: Count dividend terms

The dividend is $4z^5 - 25z^4 - 23z^3 + 11z^2 + 17z + 54$, which has 6 terms.

Step2: Set up synthetic division

Use $r=7$ (from divisor $z-7$), list coefficients: $4, -25, -23, 11, 17, 54$

$$\begin{array}{r|rrrrrr} 7 & 4 & -25 & -23 & 11 & 17 & 54 \\ \hline & & 28 & 21 & -14 & -21 & -28 \\ \hline & 4 & 3 & -2 & -3 & -4 & 26 \end{array}$$

Step3: Identify quotient and remainder

Quotient degree = dividend degree -1 = 4, coefficients: $4,3,-2,-3,-4$. Remainder is the last value.

Answer:

How many terms are in the dividend? $\boldsymbol{6}$
Quotient: $\boldsymbol{4z^4 + 3z^3 - 2z^2 - 3z - 4}$
Remainder: $\boldsymbol{26}$