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Question

X^4+4X^3+6X^2+4X+1=0

Answer 1

Step 1:

Identify Possible Rational Zeros:

It is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

The equation is .

Because the leading coefficient is , the possible rational zeros are the integer factors of the constant term .

Therefore the possible rational zeros are .

Step 2:

Consider .

Perform the synthetic substitution method by testing .

By using synthetic substitution, it can be determined that  is a rational zero.

The depressed polynomial is .

Step 3:

Consider .

Perform the synthetic substitution method on the depressed polynomial by testing .

By using synthetic substitution, it can be determined that is a rational zero.

The new depressed polynomial is .

Step 4:

Consider .

Therefore are the factors of the equation.

Step 5:

The final quotient can be written as .

Factoring the quadratic expression yields .

Zeros are .

Solution:

The zeros of the equation are .