Question

divide $5x^{4}+3x^{3}+4x^{2}+1$ by $x-2$
give your answer in the form $q(x)+\frac{r(x)}{d(x)}$
fill in the table below to complete the synthetic division.*
quotient =

Explanation:

Step1: Identify root & coefficients

Root of divisor $x-2$ is $2$. Coefficients of dividend $5x^4+3x^3+4x^2+0x+1$: $5, 3, 4, 0, 1$

Step2: Set up synthetic division table

Left box: $2$
Top row boxes: $5, 3, 4, 0, 1$

Step3: Bring down leading coefficient

Bottom left box: $5$

Step4: Multiply & add (1st column)

Multiply $5\times2=10$, add to $3$: $3+10=13$
Middle row 2nd box: $10$, bottom 2nd box: $13$

Step5: Multiply & add (2nd column)

Multiply $13\times2=26$, add to $4$: $4+26=30$
Middle row 3rd box: $26$, bottom 3rd box: $30$

Step6: Multiply & add (3rd column)

Multiply $30\times2=60$, add to $0$: $0+60=60$
Middle row 4th box: $60$, bottom 4th box: $60$

Step7: Multiply & add (4th column)

Multiply $60\times2=120$, add to $1$: $1+120=121$
Middle row 5th box: $120$, bottom 5th box: $121$

Step8: Find quotient polynomial

Bottom row (excluding last term) gives coefficients of quotient: $5,13,30,60$, so quotient is $5x^3+13x^2+30x+60$

Answer:

Completed Synthetic Division Table:

2	5	3	4	0	1
	5	13	30	60	121

Final Form:

$Q(x)+\frac{R(x)}{D(x)} = 5x^3+13x^2+30x+60+\frac{121}{x-2}$

Quotient:

$5x^3+13x^2+30x+60$