All the real zeros of the given polynomial are integers.

Victor Wall 2021-12-16 Answered
All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form.
P(x)=x42x33x2+8x4
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Expert Answer

Ben Owens
Answered 2021-12-17 Author has 27 answers
Step 1
To find zeroes of polynomial
P(x)=x42x33x2+8x4
Step 2
P(1)=0
so
x42x33x2+8x4=(x1)(x3x24x+4)
Also x=1 is zero of (x3x24x+4)
Again
x42x33x2+8x4=(x1)(x3x24x+4)
=(x1)2(x24)
±2 is zero of (x24)
x42x33x2+8x4=(x1)(x1)(x2)(x+2)

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Joseph Lewis
Answered 2021-12-18 Author has 43 answers
The given polynomial is P(x)=x42x33x2+8x4
Since the leading term is 1, any rational zero must be a divisior of the constant term 4
So the possible rational zeros are ±1,±2,±4
We test each of these possibilities
P(1)=142(1)33(1)2+8(1)4=123+84=0
P(1)=(1)42(1)33(1)2+8(1)4=1+2384=12
P(2)=242(2)33(2)2+8(2)4=161612+164=0
P(2)=(2)42(2)33(2)2+8(2)4=16+1612164=0
P(4)=442(4)33(4)2+8(4)4=25612848+324=108
P(4)=(4)42(4)33(4)2+8(4)4=256+12848324=300
The rational zeros of P are 1,2, and -2.

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