what is the reminder when 3x^2 + 6x-4 is divided by 5x+2
The correct answer and explanation is:
To find the remainder when 3×2+6x−43x^2 + 6x – 4 is divided by 5x+25x + 2, we use polynomial division or the remainder theorem for linear divisors.
🔢 Step-by-Step Polynomial Division:
We are dividing: 3×2+6x−45x+2\frac{3x^2 + 6x – 4}{5x + 2}
Let’s perform polynomial long division:
1️⃣ First Term:
Divide the leading term 3x23x^2 by the leading term 5x5x: 3x25x=35x\frac{3x^2}{5x} = \frac{3}{5}x
Multiply 35x\frac{3}{5}x by 5x+25x + 2: (35x)(5x+2)=3×2+65x\left(\frac{3}{5}x\right)(5x + 2) = 3x^2 + \frac{6}{5}x
Subtract from the original polynomial: (3×2+6x−4)−(3×2+65x)=(6x−65x)−4=245x−4(3x^2 + 6x – 4) – (3x^2 + \frac{6}{5}x) = \left(6x – \frac{6}{5}x\right) – 4 = \frac{24}{5}x – 4
2️⃣ Next Term:
Divide 245x\frac{24}{5}x by 5x5x: 245x5x=2425\frac{\frac{24}{5}x}{5x} = \frac{24}{25}
Multiply 2425\frac{24}{25} by 5x+25x + 2: 2425(5x+2)=12025x+4825=245x+4825\frac{24}{25}(5x + 2) = \frac{120}{25}x + \frac{48}{25} = \frac{24}{5}x + \frac{48}{25}
Subtract: (245x−4)−(245x+4825)=−4−4825=−10025−4825=−14825\left(\frac{24}{5}x – 4\right) – \left(\frac{24}{5}x + \frac{48}{25}\right) = -4 – \frac{48}{25} = -\frac{100}{25} – \frac{48}{25} = -\frac{148}{25}
✅ Final Answer:
The remainder is: −14825\boxed{-\frac{148}{25}}
🧠 Explanation (300 Words):
When dividing polynomials, the remainder plays a similar role to that in arithmetic. In this case, we divide the quadratic polynomial 3×2+6x−43x^2 + 6x – 4 by the linear binomial 5x+25x + 2. Since the divisor has degree 1 and the dividend has degree 2, we expect the quotient to be of degree 1 and the remainder to be a constant.
We use polynomial long division to proceed. The process mimics the steps of numeric long division but with algebraic terms. First, divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply it by the entire divisor and subtract from the original polynomial to eliminate the leading term. Repeat the process with the resulting expression.
After completing the division steps, the remaining expression is the remainder. In our case, the calculations yield a final remainder of: −14825-\frac{148}{25}
This remainder means that: 3×2+6x−4=(5x+2)(35x+2425)+(−14825)3x^2 + 6x – 4 = (5x + 2)\left(\frac{3}{5}x + \frac{24}{25}\right) + \left(-\frac{148}{25}\right)
This confirms the division result. The polynomial cannot be divided evenly by 5x+25x + 2; instead, we obtain a linear quotient and a constant (non-zero) remainder.
Understanding polynomial division is crucial in algebra and calculus, especially in simplifying expressions and finding limits in rational functions.