Certainly! Let's solve the polynomial division step by step:
We are asked to divide the polynomial [tex]\( 15x^2 + 5xy - 4y^2 \)[/tex] by the binomial [tex]\( 3x + 2y \)[/tex].
### Step 1: Setup the division
Write the polynomial [tex]\( 15x^2 + 5xy - 4y^2 \)[/tex] under the division bar, and place the divisor [tex]\( 3x + 2y \)[/tex] outside the division bar:
1. Polynomial (dividend): [tex]\( 15x^2 + 5xy - 4y^2 \)[/tex]
2. Divisor: [tex]\( 3x + 2y \)[/tex]
### Step 2: Perform the first division
Look at the first term of both the dividend and the divisor to find the first term of the quotient. We divide the first term of the dividend [tex]\( 15x^2 \)[/tex] by the first term of the divisor [tex]\( 3x \)[/tex]:
[tex]\[ \frac{15x^2}{3x} = 5x \][/tex]
So, the first term of the quotient is [tex]\( 5x \)[/tex].
### Step 3: Multiply and subtract
Multiply the entire divisor [tex]\( 3x + 2y \)[/tex] by the first term of the quotient [tex]\( 5x \)[/tex], and subtract from the original polynomial:
[tex]\[ (3x + 2y) \cdot 5x = 15x^2 + 10xy \][/tex]
Now subtract [tex]\( 15x^2 + 10xy \)[/tex] from the original polynomial:
[tex]\[
(15x^2 + 5xy - 4y^2) - (15x^2 + 10xy) = 5xy - 10xy - 4y^2 = -5xy - 4y^2
\][/tex]
### Step 4: Perform the next division
Now bring down the next term from the original polynomial (if necessary) and perform the next division. Here the next term is already included:
Divide [tex]\( -5xy \)[/tex] by [tex]\( 3x \)[/tex]:
[tex]\[ \frac{-5xy}{3x} = -\frac{5y}{3} \][/tex]
So, the next term of the quotient is [tex]\( -\frac{5y}{3} \)[/tex].
### Step 5: Multiply and subtract again
Multiply the entire divisor [tex]\( 3x + 2y \)[/tex] by [tex]\( -\frac{5y}{3} \)[/tex]:
[tex]\[ \left(3x + 2y\right) \left(-\frac{5y}{3}\right) = -5xy - \frac{10y^2}{3} \][/tex]
Subtract this result from the remaining polynomial:
[tex]\[
(-5xy - 4y^2) - (-5xy - \frac{10y^2}{3})
\][/tex]
This gives:
[tex]\[
-5xy - 4y^2 + 5xy + \frac{10y^2}{3} = -4y^2 + \frac{10y^2}{3} = -\frac{12y^2}{3} + \frac{10y^2}{3} = -\frac{2y^2}{3}
\][/tex]
Here, we get the remainder as [tex]\( -\frac{2y^2}{3} \)[/tex].
### Step 6: Consolidate the results
So, the quotient of the division is [tex]\( 5x - \frac{5y}{3} \)[/tex] and the remainder is [tex]\( -\frac{2y^2}{3} \)[/tex].
### Final Answer:
- Quotient: [tex]\( 5x - \frac{5y}{3} \)[/tex]
- Remainder: [tex]\( -\frac{2y^2}{3} \)[/tex]
Hence, we can write:
[tex]\[ 15x^2 + 5xy - 4y^2 = (3x + 2y) \left(5x - \frac{5y}{3}\right) + \left(-\frac{2y^2}{3}\right) \][/tex]
