Divide $3 x^{3} + 5 x^{2} + 2 x + 7 by x^{2} + 2 x + 3,$ find the quotient and remainder and verify the division algorithm.

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Solution

Step 1: Division
$Let x^{2} + 2 x + 3 3 x - 1 3 x^{3} + 5 x^{2} + 2 x + 7$
$\frac{3 x^{3} + 6 x^{2} + 9 x}{- x^{2} - 7 x + 7}$
$\frac{- x^{2} - 2 x - 3}{- 5 x + 10} \frac{}{}$
Step 2: Verification
By Division Algorithm
$p (x) = g (x) \times q (x) + r (x) 3 x^{3} + 5 x^{2} + 2 x + 7 = (x^{2} + 2 x + 3) (3 x - 1) + (- 5 x + 10) = 3 x^{3} - x^{2} + 6 x^{2} - 2 x + 9 x - 3 - 5 x + 10 = 3 x^{3} + 5 x^{2} + 2 x + 7 (Verified)$

Thus, quotient is $3 x - 1$ and the remainder is $- 5 x + 10 .$

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Divide 3x3+5x2+2x+7byx2+2x+3,find the quotient and remainder and verify the division algorithm.

Step 1: DivisionLetx2+2x+33x-13x3+5x2+2x+7 3x3+6x2+9x-x2-7x+7 -x2-2x-3-5x+10Step 2: VerificationBy Division Algorithmp(x)=gx×qx+rx3x3+5x2+2x+7=x2+2x+33x-1+-5x+10=3x3-x2+6x2-2x+9x-3-5x+10=3x3+5x2+2x+7(Verified)Thus, quotient is 3x-1and the remainder is -5x+10.

Divide $3 x^{3} + 5 x^{2} + 2 x + 7 by x^{2} + 2 x + 3,$ find the quotient and remainder and verify the division algorithm.