Question

use synthetic division to divide the two polynomials.
\\( y + 5 \overline{) 3y^5 + 14y^4 - 4y^3 + 7y^2 + 15y + 25} \\)
is the divisor given in (x - r) form?
how many terms are in the dividen
enter the quotient and remainder.
quotient + remainder

Explanation:

Step1: Adjust divisor to $(x-r)$ form

The divisor $y+5$ can be rewritten as $y-(-5)$, so $r=-5$.

Step2: List dividend coefficients

Dividend: $3y^5 + 14y^4 - 4y^3 + 7y^2 + 15y + 25$, coefficients are $3, 14, -4, 7, 15, 25$.

Step3: Set up synthetic division

Write $r=-5$ left, coefficients top:

$$\begin{array}{c|rrrrrr} -5 & 3 & 14 & -4 & 7 & 15 & 25 \\ \hline & & -15 & 5 & -5 & -10 & -25 \\ \hline & 3 & -1 & 1 & 2 & 5 & 0 \end{array}$$

Step4: Identify quotient and remainder

The last value is remainder. The rest are quotient coefficients, degree 1 less than dividend.

Answer:

Is the divisor given in $(x-r)$ form? Yes
How many terms are in the dividend? 6
Quotient: $3y^4 - y^3 + y^2 + 2y + 5$
Remainder: $0$