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

Kathy Williams

Kathy Williams

Answered question

2021-12-13

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)

Answer & Explanation

Fasaniu

Fasaniu

Beginner2021-12-14Added 46 answers

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=(05)(0+2)(0+3)(0d)
60=(30)(d)
d=2
Hence required p(x)=(x-5)(x+2)(x+3)(x-2)
Karen Robbins

Karen Robbins

Beginner2021-12-15Added 49 answers

P(x)=x42x319x2+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=2×88b+4c2d+60
8b+4c2d=76
-4b+2c-d=-38...(3)
0=81-27b+9c-3d+60
9b+3cd=47...(4)
21b+7c=175   16b+8c=184
3b+c=25...(5)
2b+c=-23...(6)
b=-2
6+c=25c=19
d=-137-25(-2)-5(-19)
=-137+50+95
d=-137+145
d=8
P(x)=x42x319x2+8x+60

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