If 2x4 + px3 + qx2 + x + 6 is divisible by 2x2 + x − 3, find the values for theconstants p and q.

To find the values of p and q, we can use the polynomial division method.

Step 1: Divide the first term of the dividend (2x^4) by the first term of the divisor (2x^2) to get x^2.

Step 2: Multiply the entire divisor (2x^2 + x - 3) by x^2 and subtract the result from the original polynomial.

This gives us:

2x^4 + px^3 + qx^2 + x + 6 - (2x^4 + x^3 - 3x^2) = px^3 + (q+3)x^2 - x^3 + x + 6

Simplify to get:

(px - 1)x^3 + (q + 3)x^2 + x + 6

Step 3: Repeat the process. Divide the first term of the new polynomial (px - 1)x^3 by the first term of the divisor 2x^2 to get (p/2 - 1/2)x.

Step 4: Multiply the entire divisor by (p/2 - 1/2)x and subtract the result from the new polynomial.

This gives us:

(px - 1)x^3 + (q + 3)x^2 + x + 6 - [(p - 1)x^3 + (p/2 - 1/2)x^2 - (p/2 - 1/2)x]

Simplify to get:

(q - p/2 + 2)x^2 + (1 + p/2)x + 6

Since the original polynomial is divisible by the divisor, the remainder must be zero. Therefore, the coefficients of x^2 and x in the final polynomial must be zero.

Setting these equal to zero gives us the system of equations:

q - p/2 + 2 = 0 and 1 + p/2 = 0

Solving this system gives us p = -2 and q = 1.