Final answer:
To find the value of p for which the polynomial 2x⁴ +3x³ +2px² +3x+6 is exactly divisible by (x+2), we need to use the factor theorem. Substitute x = -2 into the polynomial and solve for p: 58 + 8p = 0. Therefore, the value of p for which the polynomial is exactly divisible by (x+2) is -rac{29}{4}.
Step-by-step explanation:
To find the value of p for which the polynomial 2x⁴ +3x³ +2px² +3x+6 is exactly divisible by (x+2), we need to use the factor theorem.
The factor theorem states that if a polynomial f(x) is exactly divisible by (x-a), then f(a) = 0.
In this case, we can substitute x = -2 into the polynomial and solve for p:
- Substitute x = -2: 2(-2)⁴ +3(-2)³ +2p(-2)² +3(-2)+6 = 0
- Simplify and solve for p: 32 +24 +8p -6 -6 = 0
- Combine like terms: 32 + 24 + 8p - 6 - 6 = 0
- Solve for p: 58 + 8p = 0
- Subtract 58 from both sides: 8p = -58
- Divide both sides by 8: p = -rac{58}{8}
- Simplify: p = -rac{29}{4}
Therefore, the value of p for which the polynomial is exactly divisible by (x+2) is -rac{29}{4}.