If x4+2x3+px2+qx+9=0 is a complete square. p and q are positive. Find the value of p and q.

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belamontern9i · Accepted Answer

Step 1If it is a complete square, thenx4+2x3+px2+qx+9=(x2+ax+b)2=x4+2ax3+(2b+a2)x2+2abx+b2(we have the coefficient in front of x2 is 1 since the coefficient in front of x4 is 1). Clearly, b=3 and a=1 which gives usx4+2x3+px2+qx+9=(x2+x+3)2=x4+2x3+7x2+6x+9px2+qx=7x2+6xSo take p=7 and 6=q

Luke Kane · Accepted Answer

Step 1Okay, I&#039;m assuming you mean that the polynomialf(x)=x4+2x3+px2+qx+9 is a square, that is there is some other polynomial g(x) with g(x)×g(x)=f(x)So what could this polynomial g(x) be? Well, the highest-degree term in f(x) is x4, so that means the highest-degree term in g(x) should be x2(x2×x2=x4)So g(x)=x2+ax+b for some constants a and b. Now lets look at what happens when we multiply it out. We get f(x)=g(x)×g(x)=x4+2ax3+(2b+a2)x2+2abx+b2This tells us a few things. First of all, by looking at the x3 term we see 2a=2, or a=1. Second, by looking at the constant term, we see b2=9 so b=±3. So that gives us two possibilities for g(x).g(x)=x2+x+3 or g(x)=x2+x−3.Now lets look at what happens with p and q in both cases. Remember, by looking at the multiplication we havep=(2b+a2) and q=2ab So if p and q are positive, we have to take the +3 option. In which case p=7 and q=6.

If x^4+2x^3+px^2+qx+9=0 is a complete square. p and q are

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