Q. What is \(3x^3 – 11x^2 – 26x + 30\) divided by \(x – 5\)?

Answer

Set up synthetic division.
Use the root 5. The coefficients are 3, -11, -26, and 30.
Process the coefficients.
Bring down 3. Multiply by 5 and add: -11 + 15 = 4.

4 times 5 is 20. Add: -26 + 20 = -6.

-6 times 5 is -30. Add: 30 – 30 = 0 (remainder).
State the quotient.
\[ \frac{3x^3 – 11x^2 – 26x + 30}{x – 5} = 3x^2 + 4x – 6 \]

Detailed Explanation

Solution

Identify the synthetic root.
For divisor x – 5, use root c = 5.
Set up synthetic division.
The coefficients are 3, -11, -26, and 30.
Multiply and add.
Bring down 3. Multiply by 5 to get 15. Add to -11 to get 4.

Multiply 4 by 5 to get 20. Add to -26 to get -6.

Multiply -6 by 5 to get -30. Add to 30 to get 0 remainder.
State the quotient.
\[ 3x^2 + 4x – 6 \]

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Frequently Asked Questions

What is the quotient and remainder when dividing 3x^3 - 11x^2 - 26x + 30 by x - 5?

Quotient is 3x^2 + 4x - 6 and remainder is 0, so the division yields 3x^2 + 4x - 6 exactly.

Is x - 5 factor of 3x^3 - 11x^2 - 26x + 30?

Yes. Since the remainder is 0 when dividing by x - 5 (or evaluating the polynomial at x = 5 gives 0), x - 5 is factor.

How do I use synthetic division here?

Use root 5 with coefficients [3, -11, -26, 30]: bring down 3; multiply 3*5=15, add to get 4; multiply 4*5=20, add to get -6; multiply -6*5=-30, add to get 0. Coefficients 3,4,-6 give the quotient.

How can I check my division result is correct?

Multiply the divisor (x - 5) by the quotient (3x^2 + 4x - 6) and add the remainder (0). If you get the original polynomial, the division is correct.

How do I factor the polynomial completely over the reals?

It factors as (x - 5)(3x^2 + 4x - 6). The quadratic has roots x = (-2 ± sqrt(22))/3, so those give the other two real factors if desired.

What does the Remainder Theorem say and how is it used here?

What does the Remainder Theorem say and how is it used here?

When should I use synthetic division vs long division?

Use synthetic division when dividing by linear binomial of form x - (quick, fewer steps). Use polynomial long division for divisors that are not of that form or when coefficients/structure make synthetic inconvenient.

What is the degree of the quotient, and why?

The quotient degree is 2 because dividing degree-3 polynomial by degree-1 polynomial reduces the degree by 1 (3 - 1 = 2).

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