MATHCRAVE ALGEBRA
Polynomial Divisions
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# AI Polynomial Division Solver

## AI Polynomial Division Solver

### Polynomial Division

#### Introduction to Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps simplify expressions and solve polynomial equations.

#### Types of Polynomial Division

1. Long Division
2. Synthetic Division (for specific cases where the divisor is a binomial of the form $$x – c$$)

#### Long Division of Polynomials

Steps for Long Division:

1. Arrange Polynomials:
– Write both the dividend and the divisor in descending order of their degrees.
– For example, divide $$2x^3 + 3x^2 – 5x + 6$$ by $$x – 2$$.

– Divide the first term of the dividend by the first term of the divisor.
– $$\frac{2x^3}{x} = 2x^2$$

3. Multiply and Subtract:
– Multiply the entire divisor by the result from step 2 and subtract this from the original dividend.
– $$(2x^3 + 3x^2 – 5x + 6) – (2x^2(x – 2)) = (2x^3 + 3x^2 – 5x + 6) – (2x^3 – 4x^2) = 7x^2 – 5x + 6$$

4. Repeat:
– Repeat steps 2 and 3 with the new polynomial $$7x^2 – 5x + 6$$.
– $$\frac{7x^2}{x} = 7x$$
– Multiply and subtract: $$(7x^2 – 5x + 6) – (7x(x – 2)) = (7x^2 – 5x + 6) – (7x^2 – 14x) = 9x + 6$$

5. Continue Until Completion:
– Repeat the process until the degree of the remainder is less than the degree of the divisor.
– $$\frac{9x}{x} = 9$$
– Multiply and subtract: $$(9x + 6) – (9(x – 2)) = (9x + 6) – (9x – 18) = 24$$

6. Combine Results:
– The quotient is $$2x^2 + 7x + 9$$ and the remainder is $$24$$.
– The final result is $$2x^2 + 7x + 9 + \frac{24}{x – 2}$$.

#### Synthetic Division

When to Use:
– Synthetic division is used when the divisor is a binomial in the form $$x – c$$.

#### Steps for Synthetic Division:

1. Set Up:
– Write down the coefficients of the dividend polynomial.
– For $$2x^3 + 3x^2 – 5x + 6$$ and divisor $$x – 2$$, write $$2, 3, -5, 6$$.

2. Use the Zero of the Divisor:
– The zero of $$x – 2$$ is $$2$$.
– Write $$2$$ to the left and a vertical line to separate.

3. Bring Down the Leading Coefficient:
– Bring down the first coefficient directly below the line.
– $$2$$

– Multiply the leading coefficient by the zero of the divisor and write the result below the next coefficient, then add.
– $$2 \times 2 = 4$$
– $$3 + 4 = 7$$
– Continue this process:
– $$7 \times 2 = 14$$
– $$-5 + 14 = 9$$
– $$9 \times 2 = 18$$
– $$6 + 18 = 24$$

5. Interpret the Result:
– The result below the line represents the coefficients of the quotient polynomial.
– For the example: $$2x^2 + 7x + 9$$ and the remainder $$24$$.

6. Combine Results:
– The final result is $$2x^2 + 7x + 9 + \frac{24}{x – 2}$$.

#### Practice Problems

1. Divide $$4x^3 + 6x^2 – x + 5$$ by $$2x – 1$$ using long division.
2. Divide $$3x^4 – 2x^3 + x – 4$$ by $$x + 1$$ using synthetic division.
3. Divide $$5x^3 + 3x^2 – x – 2$$ by $$x – 3$$ using long division.