Question

Divide. $$ \left(3 a^{2}-\frac{23}{4} a-5\right) \div(4 a+3) $$

   Divide.
$$
\left(3 a^{2}-\frac{23}{4} a-5\right) \div(4 a+3)
$$

Intermediate Algebra

Intermediate Algebra

Margaret L. Lial,… 11th Edition

Chapter 5, Problem 55

Instant Answer

Solved by Expert Matthew Dunham

08/02/2020

Step 1

The divisor is \(4a+3\) and the dividend is \(3a^2-\frac{23}{4}a-5\). Show more…

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Divide. $$ \left(3 a^{2}-\frac{23}{4} a-5\right) \div(4 a+3) $$

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Key Concepts

Polynomial Division

Polynomial division is the process of dividing one polynomial by another, resulting in a quotient and a possible remainder. It mirrors the long division algorithm used in arithmetic and is based on the division algorithm for polynomials, which guarantees that for any two polynomials, there exists a unique quotient and a remainder where the degree of the remainder is less than the degree of the divisor.

Long Division of Polynomials

Long division of polynomials is a systematic method that involves dividing the leading term of the dividend by the leading term of the divisor to obtain the next term of the quotient. This term is then multiplied by the entire divisor and subtracted from the dividend, with the process repeating until the degree of the remaining polynomial is less than the degree of the divisor.

Remainder Theorem

The remainder theorem states that if a polynomial is divided by a linear divisor of the form (x - c), the remainder is equal to the polynomial evaluated at x = c. This theorem provides a quick way to determine the remainder without performing the full division, although adjustments are necessary when the divisor is not in the (x - c) form.

Algebraic Manipulation

Algebraic manipulation involves the careful handling of algebraic expressions, including combining like terms, managing fractional coefficients, and performing basic arithmetic operations. Proficiency in these skills is essential for correctly carrying out polynomial division and simplifying the results.

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