We have seen the close connection between polynomial...

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting $$ f(x)=2 x^{2}+5 x-12. $$ Use the conclusion reached in to determine whether $x-3$ is a factor of $g(x)=3 x^{3}-4 x^{2}-17 x+6 .$ If so, factor $g(x)$ completely.

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12.
$$
Use the conclusion reached in to determine whether $x-3$ is a factor of $g(x)=3 x^{3}-4 x^{2}-17 x+6 .$ If so, factor $g(x)$ completely.

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Key Concepts

Polynomial Division

Polynomial division is a process analogous to numerical long division applied to polynomials. It involves dividing a dividend polynomial by a divisor polynomial to obtain a quotient and a remainder, which can be useful for simplifying polynomial expressions and solving equations.

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This concept is fundamental in determining whether a given binomial is a factor of a polynomial and is often used in conjunction with polynomial division to factor complex polynomials completely.

Remainder Theorem

The Remainder Theorem complements the Factor Theorem by stating that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This theorem provides a quick way to evaluate the remainder without performing full polynomial division and is key to deciding if (x - c) is a factor of the polynomial.

Factoring Polynomials

Factoring polynomials is the process of expressing a polynomial as a product of simpler polynomials or factors. This technique is vital in simplifying polynomial expressions and solving polynomial equations, and it often employs methods such as polynomial division, synthetic division, and the application of the Factor and Remainder Theorems.

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