Question

Find the quotient and remainder if$3x^4-5x^3+7x-4$is divided by$x-3$.

Short answer

The quotient is $3x^3+4x^2+12x+43$and the remainder is$125$

Step-by-step solution

Step 1. Given information.

The given polynomial is $3x^4-5x^3+7x-4$and it should be divided by$x-3$.

Step 2. Dividing the given polynomial.

First, we will insert the missing $x^2$term in the dividend and divide by $x-3$.

localid="1646211602856" $$\begin{gathered} x-33x^3+4x^2+12x+433x^4-5x^3+0x^2+7x-4 \\ 3x^4-9x^3 \\ \overline{4x^3+0x^2+7x-4} \\ 4x^3-12x^2 \\ \overline{12x^2+7x-4} \\ 12x^2-36x \\ \overline{43x-4} \\ 43x-129 \\ \overline{125} \end{gathered}$$

Therefore, the quotient is$3x^3+4x^2+12x+43$and the remainder is$125$.

Step 3. Checking the result.

Multiplying quotient by divisor and then add the remainder then we get,

$$\begin{gathered} \left(x-3\right)\left(3x^3+4x^2+12x+43\right)+125 \\ =3x^4+4x^3+12x^2+43x-9x^3-12x^2-36x-129+125 \\ =3x^4-5x^3+7x-4 \end{gathered}$$

Hence this is true.