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Divide. If there is a remainder, include it as a simplified fraction.\newline(3x3+11x2+10x)÷(x+2)(3x^3 + 11x^2 + 10x) \div (x + 2)

Full solution

Q. Divide. If there is a remainder, include it as a simplified fraction.\newline(3x3+11x2+10x)÷(x+2)(3x^3 + 11x^2 + 10x) \div (x + 2)
  1. Polynomial Long Division: We will use polynomial long division to divide (3x3+11x2+10x)(3x^3 + 11x^2 + 10x) by (x+2)(x + 2). First, we divide the first term of the dividend, 3x33x^3, by the first term of the divisor, xx, to get the first term of the quotient. 3x3÷x=3x23x^3 \div x = 3x^2
  2. First Term Division: We multiply the divisor (x+2)(x + 2) by the first term of the quotient (3x2)(3x^2) and subtract the result from the dividend.\newline(3x3+11x2+10x)(3x2(x+2))=(3x3+11x2+10x)(3x3+6x2)(3x^3 + 11x^2 + 10x) - (3x^2 \cdot (x + 2)) = (3x^3 + 11x^2 + 10x) - (3x^3 + 6x^2)\newlineThis simplifies to 5x2+10x5x^2 + 10x.
  3. Subtraction and Simplification: Next, we divide the first term of the remaining polynomial, 5x25x^2, by the first term of the divisor, xx, to get the next term of the quotient.\newline5x2÷x=5x5x^2 \div x = 5x
  4. Next Term Division: We multiply the divisor (x+2)(x + 2) by the new term of the quotient (5x)(5x) and subtract the result from the remaining polynomial.(5x2+10x)(5x(x+2))=(5x2+10x)(5x2+10x)(5x^2 + 10x) - (5x \cdot (x + 2)) = (5x^2 + 10x) - (5x^2 + 10x)This simplifies to 00, so there is no remainder.

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