3x² + 6xy + y³ / x − y

Question

ScanSolve AI Tutor · Accepted Answer

To simplify the given algebraic expression, we will perform polynomial long division of the numerator $3x^2+6xy+y^3$ by the denominator $x-y$ .

Divide the leading term of the numerator ( $3x^2$ ) by the leading term of the denominator ( $x$ ).
$\frac{3x^2}{x} = 3x$
Multiply this result by the denominator:
$3x(x-y) = 3x^2 - 3xy$
Subtract this from the original numerator:
$(3x^2+6xy+y^3) - (3x^2 - 3xy) = 3x^2+6xy+y^3 - 3x^2 + 3xy = 9xy + y^3$
Divide the leading term of the new polynomial ( $9xy$ ) by the leading term of the denominator ( $x$ ).
$\frac{9xy}{x} = 9y$
Multiply this result by the denominator:
$9y(x-y) = 9xy - 9y^2$
Subtract this from the polynomial obtained in Step 1:
$(9xy+y^3) - (9xy - 9y^2) = 9xy+y^3 - 9xy + 9y^2 = y^3 + 9y^2$
The degree of the remaining polynomial ( $y^3+9y^2$ ) with respect to $x$ (which is 0) is less than the degree of the divisor ( $x-y$ ) with respect to $x$ (which is 1).
Therefore, $y^3+9y^2$ is the remainder.
The quotient is $3x+9y$ .
Write the expression in the form of Quotient + $\frac{\text{Remainder}}{\text{Divisor}}$ .
$\frac{3x^2+6xy+y^3}{x-y} = 3x+9y + \frac{y^3+9y^2}{x-y}$
We can factor out $y^2$ from the remainder term:
$y^3+9y^2 = y^2(y+9)$
So the simplified expression is:
$\boxed{3x+9y + \frac{y^2(y+9)}{x-y}}$
2 zimekamilika. Zimebaki 3 leo — tuma inayofuata.

3x² + 6xy + y³ / x - y