One zero of each polynomial is given. Use it to express the polynomial as a product of linear and irreducible quadratic factors. $$x^{3}-2 x^{2}+x-2 ; \text { zero: } x=2$$

   One zero of each polynomial is given. Use it to express the polynomial as a product of linear and irreducible quadratic factors.
$$x^{3}-2 x^{2}+x-2 ; \text { zero: } x=2$$

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 3, Problem 23

Instant Answer

Solved by Expert Montse Solis

05/09/2020

Step 1

This is because if we substitute $x=2$ into $(x-2)$, we get zero, which confirms that $x=2$ is a root of the polynomial. Show more…

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One zero of each polynomial is given. Use it to express the polynomial as a product of linear and irreducible quadratic factors. $$x^{3}-2 x^{2}+x-2 ; \text { zero: } x=2$$

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Key Concepts

Irreducible Quadratic Factors

An irreducible quadratic factor is a quadratic polynomial that cannot be factored into two linear factors over the real numbers, usually because its discriminant is negative. Recognizing and preserving these factors is essential when expressing a polynomial in its completely factored form with real coefficients.

Factor Theorem

The factor theorem is a key concept in polynomial algebra stating that if a polynomial f(x) has a value of zero at x = c, then (x - c) is a factor of the polynomial. This theorem is critical when using a known root to reduce the polynomial's degree, thereby simplifying further factorization.

Polynomial Division

Polynomial division, such as long division or synthetic division, is a process used to divide a polynomial by one of its factors, typically a linear factor. This technique helps in determining the quotient polynomial, which can be further analyzed for additional factors.

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