Q. Divide \( f(x) = 3x^3 + 8x^2 + 5x – 4 \) by \( x + 2 \)

Q: When is synthetic division appropriate instead of long division?

Synthetic division works when dividing by linear polynomial with leading coefficient 1, i.e., x - r. For x + 2 use r = -2. For other divisors or non-monic linear factors, use long division or adjust accordingly.

Answer

Set up synthetic division.
Use the root -2. The coefficients are 3, 8, 5, and -4.
Process the coefficients.
Bring down 3.

-2 times 3 = -6. Add to 8 = 2.

-2 times 2 = -4. Add to 5 = 1.

-2 times 1 = -2. Add to -4 = -6 (remainder).
State the final result.
\[ 3x^3 + 8x^2 + 5x – 4 = (x+2)(3x^2 + 2x + 1) – 6 \]

Detailed Explanation

Divide \( f(x)=3x^3+8x^2+5x-4 \) by \( x+2 \)

Identify the root of the divisor. Since the divisor is \( x+2 \), the root for synthetic division is \( r=-2 \).
Write the coefficients of \( f(x) \): they are \(3,\;8,\;5,\;-4\).
Begin synthetic division.
- Bring down the first coefficient \(3\). This becomes the leading coefficient of the quotient.
- Multiply this value by \(r=-2\): \(3\times(-2)=-6\).
- Add this to the next coefficient: \(8+(-6)=2\). This \(2\) is the next coefficient of the quotient.
- Multiply \(2\) by \(r=-2\): \(2\times(-2)=-4\).
- Add to the next coefficient: \(5+(-4)=1\). This \(1\) is the next coefficient of the quotient.
- Multiply \(1\) by \(r=-2\): \(1\times(-2)=-2\).
- Add to the last coefficient: \(-4+(-2)=-6\). This is the remainder.
Assemble the quotient and remainder.
- The quotient coefficients found are \(3,\;2,\;1\), which correspond to the polynomial \(3x^2+2x+1\).
- The remainder is \(-6\).
State the division result.
- As an identity: \(3x^3+8x^2+5x-4=(x+2)(3x^2+2x+1)+(-6)\).
- As a quotient with remainder: \(\dfrac{3x^3+8x^2+5x-4}{x+2}=3x^2+2x+1-\dfrac{6}{x+2}\).
Optional verification (expansion).
- Compute \((x+2)(3x^2+2x+1)=3x^3+8x^2+5x+2\).
- Add the remainder \(-6\): \(3x^3+8x^2+5x+2-6=3x^3+8x^2+5x-4\), which matches the original polynomial.

Final answer: Quotient is \(3x^2+2x+1\) and remainder is \(-6\).

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Frequently Asked Questions

What are the quotient and remainder when dividing f(x) = 3x^3 + 8x^2 + 5x - 4 by x + 2?

Quotient is 3x^2 + 2x + 1 and remainder is -6. So f(x) = (x + 2)(3x^2 + 2x + 1) - 6.

How can I get the remainder quickly without full division?

Use the Remainder Theorem: evaluate f at x = -2 (the root of x + 2). f(-2) = -6, which is the remainder.

How do I perform synthetic division for this problem?

Use root -2 with coefficients 3, 8, 5, -4. Bring down 3, multiply by -2 → -6, add to 8 → 2, multiply → -4, add to 5 → 1, multiply → -2, add to -4 → -6. Coefficients 3,2,1 give the quotient.

Is x + 2 factor of f(x)?

No, because the remainder is -6 (not zero). factor requires remainder 0.

What is the degree of the quotient and why?

The divisor is degree 1, original polynomial degree 3, so the quotient has degree 3 - 1 = 2.

How can I check my division result?

Multiply the divisor (x + 2) by the quotient (3x^2 + 2x + 1) and add the remainder -6. Simplify; you should recover the original polynomial.

Can the quotient 3x^2 + 2x + 1 be factored over the reals?

No. Its discriminant is 2^2 - 4*3*1 = -8 < 0, so it has no real linear factors; it factors only over complex numbers.

When is synthetic division appropriate instead of long division?