Answer:
Explanation:
You want values of p, q, r and s such that (2x + 3) (px^3 +qx^2 +rx +s) ≡ 2x^4 +13x^3 +7x^2 +18.
Find coefficients
Multiplying out the left side of the equation, we have ...
2px^4 +(3p+2q)x^3 +(2r +3q)x^2 +(2s +3r)x +3s
Match coefficients
When we match coefficients between the two polynomials, this gives rise to 5 equations in the four unknown values:
- 2p = 2
- 3p +2q = 13
- 2r +3q = 7
- 2s +3r = 0
- 3s = 18
The first and last equations tell us ...
p = 2/2 = 1
s = 18/3 = 6
Using these values in the 2nd and 4th equations makes them be ...
3 +2q = 13 ⇒ q = 10/2 = 5
12 +3r = 0 ⇒ r = -12/3 = -4
The values of p, q, r, s are ...
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Check
We can use the 3rd equation to check these results:
2r +3q = 7
2(-4) +3(5) = 7
-8 +15 = 7 . . . . . . . true