To divide the polynomial [tex]\( 2 + 2x^2 \)[/tex] by the polynomial [tex]\( x + 2 \)[/tex] and express the result in the form [tex]\( \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \)[/tex], follow these steps:
1. Identify the dividend and divisor:
- Dividend: [tex]\( 2 + 2x^2 \)[/tex]
- Divisor: [tex]\( x + 2 \)[/tex]
2. Perform polynomial division:
- Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
[tex]\[ \text{Leading term of dividend: } 2x^2 \][/tex]
[tex]\[ \text{Leading term of divisor: } x \][/tex]
[tex]\[ \frac{2x^2}{x} = 2x \][/tex]
- Multiply the divisor [tex]\( x + 2 \)[/tex] by this term [tex]\( 2x \)[/tex]:
[tex]\[ 2x \cdot (x + 2) = 2x^2 + 4x \][/tex]
- Subtract this product from the original dividend:
[tex]\[ (2 + 2x^2) - (2x^2 + 4x) = (2 + 2x^2 - 2x^2 - 4x) = -4x + 2 \][/tex]
- This leaves a new polynomial to divide: [tex]\( -4x + 2 \)[/tex]
3. Divide the new polynomial by the divisor:
- Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[ \frac{-4x}{x} = -4 \][/tex]
- Multiply the divisor [tex]\( x + 2 \)[/tex] by this term [tex]\( -4 \)[/tex]:
[tex]\[ -4 \cdot (x + 2) = -4x - 8 \][/tex]
- Subtract this product from the new polynomial:
[tex]\[ (-4x + 2) - (-4x - 8) = (-4x + 2 + 4x + 8) = 10 \][/tex]
- This leaves a remainder of [tex]\( 10 \)[/tex].
4. Express the result:
- The quotient is [tex]\( 2x - 4 \)[/tex] and the remainder is [tex]\( 10 \)[/tex].
Now, we write the final answer in the given form:
[tex]\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \][/tex]
[tex]\[ 2 + 2x^2 = (x + 2) \times (2x - 4) + 10 \][/tex]
Therefore, the quotient is [tex]\( 2x - 4 \)[/tex] and the remainder is [tex]\( 10 \)[/tex]. The division yields:
[tex]\[ 2 + 2x^2 = (x + 2)(2x - 4) + 10 \][/tex]
