Jessica J.
asked • 07/01/19Dividing the polynomial P(x) by x + 3 yields a quotient Q(x) and a remainder of 8. If Q(3)=4, find P(3) and P(−3).
P(3)= ?
P(−3)= ?
Can't figure out the answer to this problem at all. I have seen this question asked on here before, but the math was a little confusing. Can someone please explain the answer and the steps they took to reach it? Anything helps, thanks!
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1 Expert Answer
David W. answered • 07/01/19
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It is important to begin with the meaning of division and the meaning of remainder:
A/B = C with remainder D means A = B*C + D
For example, 9/2 = 4 with remainder 1. Thus, 9 = 2*4 + 1
Given: P(x) is the dividend. (x+3) is the divisor. Q(x) is the quotient. The remainder is 8.
So, using the formula above: P(x) = (x+3)*Q(x) + 8
This formula is good for any value of x. We will need to use x=3 and x=-3.
Find P(3):
P(3) = (3+3)*Q(3) + 8
P(3) = 6*Q(3) + 8
P(3) = 6*(4) + 8 [ use Q(3)=4 ]
P(3) = 32
Find P(-3):
P(-3) = (-3+3)*Q(-3) + 8
P(-3) = 0*Q(-3) + 8 [ this is why they didn't tell you the value of Q(-3) ]
P(-3) = 8
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Gene G.
I don't think there's enough information here to answer the questions. The polynomial remainder theorem tells you that P(-3) = 8. P(r) = the remainder of P(x) / (x-r). I tried synthetic division using coefficients of P as a, b, and c. You don't get numbers for results. You get expressions for the coefficients of Q and the remainder of the division, which you can set equal to 8. Using those coefficients and another synthetic division, you get expressions for a new polynomial and a remainder which you can set equal to 4. I wind up with two equations and three unknowns. Not solvable. Maybe there's a trick that I'm not seeing?07/01/19