Can I use synthetic division??

Question

Find the quotient and remainder when $2x^4+7x^3+x^2+x+4$ is divided by $2x+1$

Here coefficient of $x$ is $2$ in divisor.

Can we use here division algorithm?? Taking $x=-1/2$ and using synthetic division I got

Remainder is $3$

And quotient $2x^3+6x^2-2x+2$ Which I can simply to $x^3+3x^2-x+1$ (since $-1/2$ is a divisor of 2$) Which is the answer

Guys what is your opinion??

Answer 1

When you use synthetics division with $$x=-1/2$$ you get $$ P(x) = (x+1/2)Q(x) +R$$

Thus to get your original division you need to modify your Q(x).

$$ P(x) = (x+1/2)Q(x) +R \iff P(x) = (2x+1)Q^* (x) +R$$

Where $$Q^* (x) = (1/2) Q(x)$$

In our problem $$Q(x) = 2x^3+6x^2-2x+2$$ and $R=3$

Thus we get $$ Q^*(x) = x^3+3x^2-x+1$$ with $R=3$

Answer 2

Note that $$(2x+1)(x^3+3x^2-x+1)=2x^4+7x^3+x^2+x+1$$ so the remainder is equal to $$3$$