(x^3-8x^2+17x-10)/(x-5) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Solving (x^3 - 8x^2 + 17x - 10) / (x - 5) Using Synthetic Division

Synthetic division is a shortcut method for dividing polynomials by binomials of the form (x - a). It provides a more efficient way to find the quotient and remainder compared to long division.

Let's solve the division problem (x^3 - 8x^2 + 17x - 10) / (x - 5) using synthetic division:

Step 1: Set Up the Division

Write down the coefficients of the dividend (x^3 - 8x^2 + 17x - 10) and the divisor (x - 5). Remember to include any missing terms with a coefficient of 0.

    5 | 1  -8  17  -10
      |__________________

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient of the dividend (1) below the horizontal line.

    5 | 1  -8  17  -10
      |__________________
        1 

Step 3: Multiply and Add

Multiply the number you just brought down (1) by the divisor (5). Write the product (5) below the next coefficient of the dividend (-8). Add the two numbers (-8 + 5 = -3) and write the sum below the line.

    5 | 1  -8  17  -10
      |__________________
        1   -3

Step 4: Repeat the Process

Repeat steps 2 and 3 for the remaining coefficients.

• Multiply the last result (-3) by the divisor (5), giving -15.
• Add -15 to the next coefficient (17), resulting in 2.

    5 | 1  -8  17  -10
      |__________________
        1   -3   2

• Multiply 2 by the divisor (5), giving 10.
• Add 10 to the last coefficient (-10), resulting in 0.

    5 | 1  -8  17  -10
      |__________________
        1   -3   2    0

Step 5: Interpret the Results

The numbers below the line represent the coefficients of the quotient polynomial, starting from the highest power of x. The last number (0) is the remainder.

Therefore, the quotient is x^2 - 3x + 2 and the remainder is 0.

Final Answer:

(x^3 - 8x^2 + 17x - 10) / (x - 5) = x^2 - 3x + 2

This means that (x^3 - 8x^2 + 17x - 10) = (x - 5)(x^2 - 3x + 2).