Find whether or not the first polynomial is a factor of the second polynomial: (i) 3x + 2, 3x^4 + 5x^3 - x^2 + 13x + 10; (ii) x^2 + 1, x^4 - 3x^3 - 4x^2 + 3x + 2.

Steps to Solve

1. Identify the Division Problem

We will divide the polynomial ( 4x^2 + 7x - 11 ) by ( 2x - 4 ).

2. Set up Polynomial Long Division

Arrange the terms in decreasing order and set up the division:

$$ \begin{align*} & \quad 2x - 4 \quad , \big| \quad 4x^2 + 7x - 11 \end{align*} $$

3. Perform the First Division Step

Divide the leading term of the dividend ( 4x^2 ) by the leading term of the divisor ( 2x ):

$$ \frac{4x^2}{2x} = 2x $$

Now multiply ( 2x ) by ( 2x - 4 ):

$$ 2x(2x - 4) = 4x^2 - 8x $$

Subtract this from the original polynomial:

$$ (4x^2 + 7x - 11) - (4x^2 - 8x) = 15x - 11 $$

4. Repeat for the Next Term

Now divide the new leading term ( 15x ) by ( 2x ):

$$ \frac{15x}{2x} = \frac{15}{2} $$

Multiply ( \frac{15}{2} ) by the entire divisor:

$$ \frac{15}{2}(2x - 4) = 15x - 30 $$

Subtract this from the new dividend:

$$ (15x - 11) - (15x - 30) = 19 $$

5. Summarize Results

The final quotient is:

$$ 2x + \frac{15}{2} $$

And the remainder is:

$$ 19 $$