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?

The Remainder Theorem: dividing f(x) by x - gives remainder f(a). Here f(5)=0, so remainder is 0 and x - 5 is factor.

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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