Reviewed by Franz Jerby Delos Santos
Use synthetic division to solve (x³ + 1) ÷ (x – 1). What is the quotient?
Answer: The quotient of (x³ + 1) ÷ (x – 1) using synthetic division is x² + x + 1
Synthetic division is a shortcut method specifically used to divide polynomials when the divisor is a linear binomial of the form (x – c). Here, we use synthetic division to divide x³ + 1 by x – 1, step-by-step, to find the quotient polynomial.
Methods
Math Tutor Explanation Using Synthetic Division:
Synthetic division simplifies the process of dividing a polynomial by a linear factor by focusing only on the coefficients.
Step 1: Step 1: Set up the synthetic division using the zero of x – 1, which is 1, and the coefficients of x³ + 1 (which are 1, 0, 0, 1)
Step 2: Step 2: Bring down the leading coefficient (1) and multiply by the zero, successively add products to the next coefficient
Math Tutor Explanation Using Long Division (for verification):
Long division can also be used to divide the polynomials and confirm the quotient found by synthetic division.
Step 1: Step 1: Divide the highest degree term of the dividend (x³) by the highest degree term of the divisor (x), getting x²
Step 2: Step 2: Multiply the entire divisor (x – 1) by x² and subtract the result from the dividend, repeat with the new polynomial
Step 1:
Step 2:
Math Tutor suggests: More Practice with Synthetic Division and Polynomials
Continue mastering synthetic division and polynomial division with these related exercises and explanations.
FAQ on Synthetic Division of Polynomials
Can synthetic division be used if the divisor isn’t linear?
No, synthetic division only works directly when the divisor is linear and of the form (x – c).
What do I do if a term is missing in the polynomial?
Insert a zero coefficient for the missing term to maintain proper order of powers.
What is the remainder when dividing x³ + 1 by x – 1?
The remainder is 2.
Does synthetic division give the same answer as polynomial long division?
Yes, both methods will yield the same quotient and remainder.
Why is synthetic division preferred over long division?
Synthetic division is faster and involves fewer steps when the divisor is a linear binomial.
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