Question

Find the quotient and remainder when t⁴ −t² −3t−7 is divided by t² −3t+8.

A) Quotient: t² −4, Remainder: 11t−71
B) Quotient: t² −2t+9, Remainder: −43
C) Quotient: t² −3t+8, Remainder: −7
D) Quotient: t² −3t+6, Remainder: 1

Answer 1

The quotient of the division of t⁴ − t² − 3t − 7 by t² − 3t + 8 is t² − 2t + 9, and the remainder is − 43, which corresponds to option B.

Step-by-step explanation:

We need to find the quotient and remainder when dividing t⁴ − t² − 3t − 7 by t² − 3t + 8. Performing the polynomial long division step-by-step, we obtain:

• Divide the first term of the numerator, t⁴, by the first term of the denominator, t², to get t² as the first term of the quotient.
• Multiply the entire divisor t² − 3t + 8 by the first term of the quotient t², and subtract the resulting polynomial from the original numerator.
• Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

The quotient obtained from the division is t² − 2t + 9, and the remainder is − 43. Therefore, the correct quotient and remainder for this division are t² − 2t + 9 and − 43, respectively, which corresponds to option B.