Find the quotient and remainder using long division.

The quotient is
The remainder is

Ask by Mcfarlane Li. in the United States

Mar 15,2025

Question

Find the quotient and remainder using long division. $$ \frac{3 x^{3}+5 x^{2}-24 x+16}{3 x-4} $$ The quotient is The remainder is

UpStudy AI · Accepted Answer

1. **Divide the leading terms**:  
   Divide $$\displaystyle 3x^3$$ by $$\displaystyle 3x$$:  
   $$
   \frac{3x^3}{3x} = x^2
   $$  
   This $$x^2$$ is the first term of the quotient.

2. **Multiply and subtract**:  
   Multiply the divisor by $$x^2$$:  
   $$
   x^2(3x-4) = 3x^3 - 4x^2
   $$  
   Subtract this from the original polynomial:  
   $$
   (3x^3+5x^2) - (3x^3-4x^2) = 9x^2
   $$  
   Bring down the remaining terms:
   $$
   9x^2 - 24x + 16
   $$

3. **Second division step**:  
   Divide the leading term of the new polynomial by $$3x$$:  
   $$
   \frac{9x^2}{3x} = 3x
   $$  
   Multiply the divisor by $$3x$$:  
   $$
   3x(3x-4) = 9x^2 - 12x
   $$  
   Subtract:
   $$
   (9x^2-24x) - (9x^2-12x) = -12x
   $$  
   Bring down the constant to obtain:
   $$
   -12x + 16
   $$

4. **Third division step**:  
   Divide $$-12x$$ by $$3x$$:  
   $$
   \frac{-12x}{3x} = -4
   $$  
   Multiply the divisor by $$-4$$:  
   $$
   -4(3x-4) = -12x + 16
   $$  
   Subtract:
   $$
   (-12x+16) - (-12x+16) = 0
   $$

5. **Conclusion**:  
   The quotient is:
   $$
   x^2+3x-4
   $$
   and the remainder is:
   $$
   0
   $$
The quotient is $$ x^2 + 3x - 4 $$ and the remainder is 0.

Find the quotient and remainder using long division.

The quotient is
The remainder is

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Find the quotient and remainder using long division. The quotient is The remainder is