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Here’s a step-by-step solution with detailed explanations to find the values of p and q:
1. Factor the divisor:
- We start by factoring the divisor polynomial x^2 + 7x + 12 to find its roots.
- Factoring it gives us (x + 4)(x + 3).
2. Apply the factor theorem:
- The factor theorem states that if a polynomial f(x) is divisible by (x – a), then f(a) = 0.
- Since x^4 + 7x^3 + 7x^2 + px + q is divisible by both (x + 4) and (x + 3), we can substitute x = -4 and x = -3 into the polynomial and equate the results to 0.
3. Substitute x = -3:
- (-3)^4 + 7(-3)^3 + 7(-3)^2 + p(-3) + q = 0
- 81 – 189 + 63 – 3p + q = 0
- -45 – 3p + q = 0 (Equation 1)
4. Substitute x = -4:
- (-4)^4 + 7(-4)^3 + 7(-4)^2 + p(-4) + q = 0
- 256 – 448 + 112 – 4p + q = 0
- -80 – 4p + q = 0 (Equation 2)
5. Solve the system of equations:
- Now we have two equations with two unknowns, p and q.
- Subtracting Equation 2 from Equation 1, we get:
- Substituting p = -35 into Equation 1 or 2, we get:
Finally, substitute the values of p and q in the given polynomial.
Therefore, the values of p and q are p = -35 and q = -60.
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