Write a polynomial, P(x), in factored form given the following requirements. Degree: 4 Leading coefficient 1 Zeros at (5,0), (−2,0) and (−3,0) y-intercept at (0,60)

Question

Write a polynomial, P(x), in factored form given the following requirements.  Degree: 4  Leading coefficient 1  Zeros at (5,0), (−2,0) and (−3,0)  y-intercept at (0,60)

Fasaniu · Accepted Answer

Step 1  Given, p(x) has  Degree: 4  Leading coefficient 1  Zeros at (5,0), (−2,0) and (−3,0)  y-intercept at (0,60)  Step 2  p(x) is a 4 degree polynomial so it has 4 roots. We can assume roots are a,b,c and d  So that p(x) can be written as  p(x)=(x-a)(x-b)(x-c)(x-d)  Also given that p(x) has zeros at (5,0), (−2,0) and (−3,0)  Hence a=5, b=-2 and c=-3  Hence p(x)=(x-5)(x+2)(x+3)(x-d)  The y-intercept is the point where the function has an input value of zero.  ⇒p(0)=60=(0−5)(0+2)(0+3)(0−d)  ⇒60=(−30)(−d)  ⇒d=2  Hence required p(x)=(x-5)(x+2)(x+3)(x-2)

Karen Robbins · Accepted Answer

P(x)=x4−2x3−19x2+8x+60  Let p(x)=ax4+bx3+cx2+dx+e  p(x)=x4+bx3+cx2+dx+e  Now p(0)=0+e  60=e  Now p(x)=x4+bx3+cx2+dx+60...(1)  0=625+125b+25c+5d+60  ⇒125b+25c+5d=−685  25b+5c+d=-137...(2)  0=2×8−8b+4c−2d+60  ⇒−8b+4c−2d=−76  -4b+2c-d=-38...(3)  0=81-27b+9c-3d+60  ⇒−9b+3c−d=−47...(4)  21b+7c=−175   16b+8c=−184  ⇒3b+c=−25...(5)  2b+c=-23...(6)  b=-2  −6+c=−25⇒c=−19  d=-137-25(-2)-5(-19)  =-137+50+95  d=-137+145  d=8  P(x)=x4−2x3−19x2+8x+60

Write a polynomial, P(x), in factored form given the following

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