Questions: Divide. (6 x^2 + 17 x + 15) ÷ (2 x + 3) Your answer should give the quotient and the Quotient: Remainder:

Transcript text: Divide. \[ \left(6 x^{2}+17 x+15\right) \div(2 x+3) \] Your answer should give the quotient and the Quotient: $\square$ Remainder: $\square$

Solution

Solution Steps

To divide the polynomial $6x^2 + 17x + 15$ by $2x + 3$, we can use polynomial long division. The process involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, subtracting from the original polynomial, and repeating the process with the remainder until the degree of the remainder is less than the degree of the divisor.

Step 1: Define the Polynomials

We start with the dividend $6x^2 + 17x + 15$ and the divisor $2x + 3$.

Step 2: Perform Polynomial Division

Using polynomial long division, we divide $6x^2 + 17x + 15$ by $2x + 3$.

Divide the leading term of the dividend $6x^2$ by the leading term of the divisor $2x$: \[ \frac{6x^2}{2x} = 3x \]
Multiply the entire divisor $2x + 3$ by $3x$: \[ 3x(2x + 3) = 6x^2 + 9x \]
Subtract this result from the original polynomial: \[ (6x^2 + 17x + 15) - (6x^2 + 9x) = 8x + 15 \]
Now, divide the leading term $8x$ by the leading term of the divisor $2x$: \[ \frac{8x}{2x} = 4 \]
Multiply the entire divisor $2x + 3$ by $4$: \[ 4(2x + 3) = 8x + 12 \]
Subtract this from the current remainder: \[ (8x + 15) - (8x + 12) = 3 \]

Step 3: Identify the Quotient and Remainder

The quotient from the division is $3x + 4$ and the remainder is $3$.