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Use the long division method to find the result when [tex]8x^4 - 22x^3 + 23x^2 + x + 13[/tex] is divided by [tex]2x^2 - 5x + 4[/tex]. If there is a remainder, express the...

Answer :

Final answer:

To divide the polynomial 8x⁴ - 22x³ + 23x² + x + 13 by 2x² - 5x + 4 using long division, we obtain the quotient 4x² - x and the remainder 2x² - 3x + 13.

Explanation:

To divide the polynomial 8x⁴ - 22x³ + 23x² + x + 13 by 2x² - 5x + 4, we use the long division method.

We begin by dividing the first term of the dividend, 8x⁴, by the first term of the divisor, 2x². This gives us 4x² as the first term of the quotient.

Then, we multiply the entire divisor, 2x² - 5x + 4, by 4x² to obtain 8x⁴ - 20x³ + 16x².

We subtract this product from the dividend, 8x⁴ - 22x³ + 23x² + x + 13, and we get -2x³ + 7x² + x + 13.

Next, we divide the first term of the resulting polynomial, -2x³, by the first term of the divisor, 2x². This gives us -x as the next term of the quotient.

Then, we multiply the entire divisor, 2x² - 5x + 4, by -x to obtain -2x³ + 5x² - 4x.

Subtracting this product from the previous result, -2x³ + 7x² + x + 13, we get 2x² - 3x + 13.

Since the degree of the resulting polynomial, 2x² - 3x + 13, is less than the degree of the divisor, we cannot continue dividing. Therefore, the quotient is 4x² - x and the remainder is 2x² - 3x + 13.

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