Divide using synthetic division. (x^4-3 x+1) ÷(x+5) |...

Name: Problem 78, Chapter 4: Algebra and Trigonometry Real Mathematics, Real People
Uploaded: 2020-08-04T06:34:51-08:00
Duration: 3 min 14 s
Channel: Ankit Gupta
Description: Divide using synthetic division. (x^4-3 x+1) ÷(x+5)

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Key Concepts

Synthetic Division

Synthetic division is a streamlined method of performing polynomial division, specifically when dividing by a linear factor of the form (x - c). Instead of working with the long division algorithm involving variables and exponents, synthetic division focuses on the coefficients, making the process quicker and less error prone. It involves a series of multiplication and addition steps that yield the coefficients of the quotient and the remainder.

Dividing by a Linear Factor

When dividing a polynomial by a linear factor, such as (x + 5) which can be rewritten as (x - (-5)), the synthetic division method allows you to test the value c = -5 directly. This complies with the Remainder Theorem, which states that the remainder of such a division is the value of the polynomial evaluated at x = c.

Including Zero Coefficients for Missing Terms

In synthetic division, all powers of x must be accounted for in the dividend. If any terms are missing (i.e., if a certain power of x is absent), a zero must be used as its coefficient. This ensures that each step of the synthetic division aligns correctly with the descending powers of x and that the final quotient is constructed accurately.

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