Questions: Divide the following polynomials: (x^3 - 16x^2 + 73x - 90) ÷ (x^2 - 7x + 10)

Divide the following polynomials: (x^3 - 16x^2 + 73x - 90) ÷ (x^2 - 7x + 10)
Transcript text: Instructions: Choose an answer and hit 'next'. You will receive your score and answers at the end. Divide the following polynomials: \[ \left(x^{3}-16 x^{2}+73 x-90\right) \div\left(x^{2}-7 x+10\right) \]
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Solution

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Solution Steps

Step 1: Polynomial Division Setup

We are tasked with dividing the polynomial \( x^{3} - 16 x^{2} + 73 x - 90 \) by \( x^{2} - 7 x + 10 \).

Step 2: First Division

We divide the leading term of the dividend \( x^{3} \) by the leading term of the divisor \( x^{2} \): \[ \frac{x^{3}}{x^{2}} = x \] Next, we multiply the entire divisor \( x^{2} - 7 x + 10 \) by \( x \): \[ x \cdot (x^{2} - 7 x + 10) = x^{3} - 7 x^{2} + 10 x \] We subtract this from the original polynomial: \[ (x^{3} - 16 x^{2} + 73 x - 90) - (x^{3} - 7 x^{2} + 10 x) = -9 x^{2} + 63 x - 90 \]

Step 3: Second Division

Now, we divide the leading term of the new polynomial \( -9 x^{2} \) by the leading term of the divisor \( x^{2} \): \[ \frac{-9 x^{2}}{x^{2}} = -9 \] We multiply the entire divisor \( x^{2} - 7 x + 10 \) by \( -9 \): \[ -9 \cdot (x^{2} - 7 x + 10) = -9 x^{2} + 63 x - 90 \] Subtracting this from our current polynomial gives us: \[ (-9 x^{2} + 63 x - 90) - (-9 x^{2} + 63 x - 90) = 0 \]

Final Answer

The quotient of the division is \( x - 9 \) and the remainder is \( 0 \). Therefore, we can express the result of the division as: \[ \frac{x^{3} - 16 x^{2} + 73 x - 90}{x^{2} - 7 x + 10} = x - 9 \] Thus, the final answer is: \[ \boxed{x - 9} \]

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