Question

divide.

\\((8x^3 + 22x^2 + 19x + 2) \div (4x^2 + 5x)\\)

your answer should give the quotient and the remainder.

quotient:
remainder:

Explanation:

Set up the polynomial division

We divide the polynomial \(8x^3 + 22x^2 + 19x + 2\) by \(4x^2 + 5x\).

Divide the leading terms

Divide the leading term of the dividend by the leading term of the divisor:

$$ \frac{8x^3}{4x^2} = 2x $$

This is the first term of our quotient.

Multiply and subtract

Multiply \(2x\) by the divisor \(4x^2 + 5x\):

$$ 2x(4x^2 + 5x) = 8x^3 + 10x^2 $$

Subtract this from the dividend:

$$ (8x^3 + 22x^2 + 19x + 2) - (8x^3 + 10x^2) = 12x^2 + 19x + 2 $$

Divide the new leading terms

Divide the leading term of the remaining polynomial by the leading term of the divisor:

$$ \frac{12x^2}{4x^2} = 3 $$

This is the second term of our quotient.

Multiply and subtract again

Multiply \(3\) by the divisor \(4x^2 + 5x\):

$$ 3(4x^2 + 5x) = 12x^2 + 15x $$

Subtract this from the remaining polynomial:

$$ (12x^2 + 19x + 2) - (12x^2 + 15x) = 4x + 2 $$

Since the degree of \(4x + 2\) is less than the degree of \(4x^2 + 5x\), the division is complete.

Answer:

Quotient: \(2x + 3\)

Remainder: \(4x + 2\)