Answer :
Final Answer:
(5x³ - 7x² + 0x - 2)
We used synthetic division to divide the given polynomials, resulting in the quotient (5x³ - 7x² + 0x - 2).
Explanation:
To perform synthetic division of the given polynomials, we first need to set up the problem properly. Write both the dividend, 10x⁶ - 14x⁴ + x³ - 19x² + 4, and the divisor, 2x³ + 2x - 1, in descending order of exponents, ensuring that all exponents are represented, even if the coefficient is zero. Then, we'll focus on the leading terms, which are 10x⁶ and 2x³.
Divide the leading term of the dividend by the leading term of the divisor: (10x⁶ / 2x³) = 5x³. = 5x³. This is the first term of the quotient. Now, we multiply the divisor, 2x³ + 2x - 1, by the quotient term, 5x³, and subtract this result from the dividend:
(10x⁶ - 14x⁴ + x³ - 19x² + 4) - (5x³ * (2x³ + 2x - 1))
This simplifies to:
5x³(2x³ + 2x - 1) = 10x⁶ + 10x⁴ - 5x³
Now, subtract this from the original dividend:
(10x⁶ - 14x⁴ + x³ - 19x² + 4) - (10x⁶ + 10x⁴ - 5x³)
After subtraction, you'll get:
(-14x⁴ + x³ - 19x² + 4) - (10x⁴ - 5x³)
Now, we focus on the new leading terms, which are -14x⁴ and -10x⁴. Divide them: (-14x⁴ / -10x⁴) = 7/5.
So, the next term in the quotient is 7/5, and we repeat the process. Continue this process until you've divided all terms. The final answer is (5x³ - 7x² + 0x - 2).
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