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Chapter 2: Problem 77

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{4}-4 x^{3}+8 x^{2}-16 x+16$$

Short Answer

Expert verified
The polynomial and the product of linear factors is \((x - 2)^4 \). The only zero of the function is '2', with a multiplicity of four.

Step by step solution

01

Hypothesize a potential root

We guess '1' as a potential root considering that the coefficients sums up to zero. This is based on the Rational Root theorem
02

Confirm the root using synthetic division

Carrying out synthetic division with '1', gives us another polynomial of degree three \(x^3 - 3x^2 + 5x - 16\). The remainder is not zero, so '1' is not a root.
03

Refine the root guess

Since '1' was close but not exact, we try with '2'. Synthetic division with '2' gives \( x^3 - 2x^2 + 4x - 8 \) with a remainder of zero. Thus, '2' is a root of the given polynomial.
04

Factoring the polynomial using the found root

Factoring the polynomial using x-2, we get \((x - 2)^4 \) (note that this is a fourth degree polynomial, one of the factors is repeated four times). This means '2' is a zero of multiplicity 4.
05

Express the polynomial as product of linear factors

Polynomial written as a product of linear factors hence will be \((x - 2)^4 \)

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