Divide the polynomial by the linear factor with synthetic...

Divide the polynomial by the linear factor with synthetic division. Indicate the quotient $Q(x)$ and the remainder $r(x)$.
$$\left(2 x^{4}-3 x^{3}+7 x^{2}-4\right) \div\left(x-\frac{2}{3}\right)$$

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Key Concepts

Remainder Theorem

The Remainder Theorem states that if a polynomial f(x) is divided by a linear divisor of the form (x - c), then the remainder of the division is equal to f(c). This theorem provides a useful shortcut for finding the remainder without performing the complete division process.

Synthetic Division

Synthetic division is a simplified technique for dividing a polynomial by a linear binomial of the form (x - c). It streamlines the long division process by using only the coefficients of the polynomial and a few arithmetic operations, which makes it a quick method for finding both the quotient and the remainder.

Polynomial Division

Polynomial division is a process analogous to numerical long division but applied to polynomials. It involves determining a quotient polynomial and a remainder polynomial such that the original polynomial equals the product of the divisor polynomial and the quotient plus the remainder. This division is guaranteed by the Division Algorithm for Polynomials.

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