All the real zeros of the given polynomial are integers.

Ernest Ryland 2021-12-15 Answered
All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form.
P(x)=x3+3x24
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Expert Answer

jean2098
Answered 2021-12-16 Author has 38 answers
Step 1
Given: All the real zeros of the given polynomial are integers.
Find the zeros, and write the polynomial in factored form.
P(x)=x3+3x24
Step 2
To Find the zeros, and write the polynomial in factored form:
P(x)=x3+3x24
=(x1)(x+2)2
=(x-1)(x+2)(x+2)
Therefore, The zeros of P(x) is P(x)=0
Hence, (x-1)(x+2)(x+2)=0
x=1,2,2

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Lynne Trussell
Answered 2021-12-17 Author has 32 answers
Consider the polynomial function in the textbook.
P(x)=x3+3x24
The objective of the questions is to find the zeros and write the polynomial in factored form.
Consider P(x)=x3+3x24,
The leading coefficient is 1 and the constant term is -4, any rational zero must be a divisior of the constant term -4.
So, the possible rational zeros are ±1,±2 and ±4
Test each of these possibilities.
Let x=1
P(x)=x3+3x24
P(1)=(1)3+3(1)24
=1+3-4
=0
Now test for x=-1
P(x)=x3+3x24
P(1)=(1)3+3(1)4
=-1+3-4
=-2
Similarly, for x=2
P(x)=x3+3x24
P(2)=(2)3+3(2)24
=8+12-4
=16
Similarly, for x=-2
P(x)=x3+3x24
P(2)=(2)3+3(2)24
=-8+12-4
=0
Similarly, for x=4
P(x)=x3+3x24
P(4)=43+3(4)24
=64+48-4
=108
Similarly, for x=-4
P(x)=x3+3x24
P(4)=(4)3+3(4)24
=-64-48-4
=-116
Hence, real zeros of P(x)=x3+3x24 are 1 and -2.
Now find the factored form of P(x)=x3+3x24,
P(x)=x3+3x24
P(x)=(x+2)2(x1)
Hence, the factored form of P(x)=x3+3x24 is (x+2)2(x1).

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