Question

Use synthetic division to divide. $$ x^{2}+4 x-21 $$

   Use synthetic division to divide.
$$
x^{2}+4 x-21
$$

Algebra for College Students

Algebra for College Students

Margaret L. Lial,… 9th Edition

Chapter 11, Problem 10

Instant Answer

Solved by Expert Aman Gupta

02/24/2022

Step 1

Write the coefficients of the polynomial $x^{2}+4x-21$ in a row. The coefficients are 1, 4, and -21. Write the root of the divisor $x-3$ on the left. The root is 3. \[ \begin{array}{c|ccc} 3 & 1 & 4 & -21 \\ \end{array} \] Show more…

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Use synthetic division to divide. $$ x^{2}+4 x-21 $$

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Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). Instead of writing out long division with variables and powers, synthetic division uses only the coefficients of the polynomial and a straightforward iterative process to obtain the quotient and remainder quickly, making it efficient for polynomials of higher degree as well.

Polynomial Division

Polynomial division is the process of dividing one polynomial (the dividend) by another (the divisor) to obtain a quotient and possibly a remainder. The process is analogous to numerical long division and relies on aligning like terms and subtracting multiples of the divisor to reduce the dividend step by step.

Division Algorithm for Polynomials

The division algorithm for polynomials states that for any two polynomials, where the divisor is nonzero, there exist unique quotient and remainder polynomials such that the original polynomial (dividend) equals the divisor multiplied by the quotient plus the remainder. The degree of the remainder is always less than the degree of the divisor.

Factor Theorem

The factor theorem is a specialized result in algebra that provides a link between the factors of a polynomial and its zeros. It states that a polynomial has a factor (x - c) if and only if the polynomial evaluates to zero at x = c. This theorem is often used in tandem with synthetic division to both factor polynomials and locate their roots.

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