Find the quotient and the remainder (if any), when:

3x⁴ + 6x³ - 6x² + 2x - 7 is divided by x - 3.

Dividing 3x⁴ + 6x³ - 6x² + 2x - 7 by x - 3

$\begin{array}{l} \phantom{x - 3)}{3x^3 + 15x^2 + 39x + 119} \ x - 3\overline{\smash{\big)}3x^4 + 6x^3 - 6x^2 + 2x - 7} \ \phantom{x - 3}\underline{\underset{-}{}3x^4 \underset{+}{-}9x^3} \ \phantom{{x - 3)}{3x^4567}}15x^3 - 6x^2 + 2x - 7 \ \phantom{{x - 3}{3x^456}}\underline{\underset{-}{+}15x^3 \underset{+}{-} 45x^2} \ \phantom{{x - 3}{3x^456 + 15x^3 + }}39x^2 + 2x - 7 \ \phantom{{x - 3}{3x^456 + 15x^3}}\underline{\underset{-}{+}39x^2 \underset{+}{-} 117x} \ \phantom{{x - 3}{3x^456 + 15x^3 + 39x^2 +}}119x - 7 \ \phantom{{x - 3}{3x^456 + 15x^3 + 39x^2}}\underline{\underset{-}{+}119x \underset{+}{-} 357} \ \phantom{{x - 3}{3x^456 + 15x^3 + 39x^2 + 117x +}}350 \end{array}$

Quotient = 3x³ + 15x² + 39x + 119

Remainder = 350

Verification:

Quotient x Divisor + Remainder

= (3x³ + 15x² + 39x + 119) $\times$ (x - 3) + 350

= x $\times$ (3x³ + 15x² + 39x + 119) - 3 $\times$ (3x³ + 15x² + 39x + 119) + 350

= 3x⁽³⁺¹⁾ + 15x⁽²⁺¹⁾ + 39x⁽¹⁺¹⁾ + 119x - 9x³ - 45x² - 117x - 357 + 350

= 3x⁴ + 15x³ + 39x² + 119x - 9x³ - 45x² - 117x - 357 + 350

= 3x⁴ + (15x³ - 9x³) + (39x² - 45x²) + (119x - 117x) + (- 357 + 350)

= 3x⁴ + 6x³ - 6x²+ 2x - 7

= Dividend