Question
Verify the division algorithm for the polynomials $p(x) = 2x^4 - 6x^3 + 2x^2 - x + 2$ and $g(x) = x + 2.$
✓
Answer
$p(x) = 2x^4 - 6x^3 + 2x^2 - x + 2$ and $g(x) = x + 2$
Quotient $= 2x^3 - 10x^2 + 22x - 45$
Remainder $= 92$
Verification: Divisor $\times$ Quotient + Remainder
$= (x + 2) \times (2x^3 - 10x^2 + 22x - 45) + 92$
$= x (2x^3 - 10x^2 + 22x - 45) + 2(2x^3 - 10x^2 + 22x - 45) + 92$
$= 2x^4 - 10x^3 + 22x^2 - 45x + 4x^3 - 20x^2 + 44x - 90 + 92$
$= 2x^4 - 6x^3 + 2x^2 - x + 2$
= Dividend Hence verified.
Quotient $= 2x^3 - 10x^2 + 22x - 45$
Remainder $= 92$
Verification: Divisor $\times$ Quotient + Remainder
$= (x + 2) \times (2x^3 - 10x^2 + 22x - 45) + 92$
$= x (2x^3 - 10x^2 + 22x - 45) + 2(2x^3 - 10x^2 + 22x - 45) + 92$
$= 2x^4 - 10x^3 + 22x^2 - 45x + 4x^3 - 20x^2 + 44x - 90 + 92$
$= 2x^4 - 6x^3 + 2x^2 - x + 2$
= Dividend Hence verified.
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