Solve for y (1-y)((2-y)(-3-y))-24+12y=0

$(1 - y) ((2 - y) (- 3 - y)) - 24 + 12 y = 0$

Step 1

Simplify $(1 - y) (2 - y) (- 3 - y) - 24 + 12 y$ .

Simplify each term.

Expand $(1 - y) (2 - y)$ using the FOIL Method.

Apply the distributive property.

$(1 (2 - y) - y (2 - y)) (- 3 - y) - 24 + 12 y = 0$

Apply the distributive property.

$(1 \cdot 2 + 1 (- y) - y (2 - y)) (- 3 - y) - 24 + 12 y = 0$

Apply the distributive property.

$(1 \cdot 2 + 1 (- y) - y \cdot 2 - y (- y)) (- 3 - y) - 24 + 12 y = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$(2 + 1 (- y) - y \cdot 2 - y (- y)) (- 3 - y) - 24 + 12 y = 0$

Multiply $- y$ by $1$ .

$(2 - y - y \cdot 2 - y (- y)) (- 3 - y) - 24 + 12 y = 0$

Multiply $2$ by $- 1$ .

$(2 - y - 2 y - y (- y)) (- 3 - y) - 24 + 12 y = 0$

Rewrite using the commutative property of multiplication.

$(2 - y - 2 y - 1 \cdot - 1 y \cdot y) (- 3 - y) - 24 + 12 y = 0$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$(2 - y - 2 y - 1 \cdot - 1 (y \cdot y)) (- 3 - y) - 24 + 12 y = 0$

Multiply $y$ by $y$ .

$(2 - y - 2 y - 1 \cdot - 1 y^{2}) (- 3 - y) - 24 + 12 y = 0$

Multiply $- 1$ by $- 1$ .

$(2 - y - 2 y + 1 y^{2}) (- 3 - y) - 24 + 12 y = 0$

Multiply $y^{2}$ by $1$ .

$(2 - y - 2 y + y^{2}) (- 3 - y) - 24 + 12 y = 0$

Subtract $2 y$ from $- y$ .

$(2 - 3 y + y^{2}) (- 3 - y) - 24 + 12 y = 0$

Expand $(2 - 3 y + y^{2}) (- 3 - y)$ by multiplying each term in the first expression by each term in the second expression.

$2 \cdot - 3 + 2 (- y) - 3 y \cdot - 3 - 3 y (- y) + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Simplify each term.

Multiply $2$ by $- 3$ .

$- 6 + 2 (- y) - 3 y \cdot - 3 - 3 y (- y) + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Multiply $- 1$ by $2$ .

$- 6 - 2 y - 3 y \cdot - 3 - 3 y (- y) + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Multiply $- 3$ by $- 3$ .

$- 6 - 2 y + 9 y - 3 y (- y) + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Rewrite using the commutative property of multiplication.

$- 6 - 2 y + 9 y - 3 \cdot - 1 y \cdot y + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$- 6 - 2 y + 9 y - 3 \cdot - 1 (y \cdot y) + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Multiply $y$ by $y$ .

$- 6 - 2 y + 9 y - 3 \cdot - 1 y^{2} + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Multiply $- 3$ by $- 1$ .

$- 6 - 2 y + 9 y + 3 y^{2} + y^{2} \cdot - 3 + y^{2} (- y) - 24 + 12 y = 0$

Move $- 3$ to the left of $y^{2}$ .

$- 6 - 2 y + 9 y + 3 y^{2} - 3 \cdot y^{2} + y^{2} (- y) - 24 + 12 y = 0$

Rewrite using the commutative property of multiplication.

$- 6 - 2 y + 9 y + 3 y^{2} - 3 y^{2} - y^{2} y - 24 + 12 y = 0$

Multiply $y^{2}$ by $y$ by adding the exponents.

Move $y$ .

$- 6 - 2 y + 9 y + 3 y^{2} - 3 y^{2} - (y \cdot y^{2}) - 24 + 12 y = 0$

Multiply $y$ by $y^{2}$ .

Raise $y$ to the power of $1$ .

$- 6 - 2 y + 9 y + 3 y^{2} - 3 y^{2} - (y^{1} y^{2}) - 24 + 12 y = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 6 - 2 y + 9 y + 3 y^{2} - 3 y^{2} - y^{1 + 2} - 24 + 12 y = 0$

Add $1$ and $2$ .

$- 6 - 2 y + 9 y + 3 y^{2} - 3 y^{2} - y^{3} - 24 + 12 y = 0$

Combine the opposite terms in $- 6 - 2 y + 9 y + 3 y^{2} - 3 y^{2} - y^{3}$ .

Subtract $3 y^{2}$ from $3 y^{2}$ .

$- 6 - 2 y + 9 y + 0 - y^{3} - 24 + 12 y = 0$

Add $- 6 - 2 y + 9 y$ and $0$ .

$- 6 - 2 y + 9 y - y^{3} - 24 + 12 y = 0$

Add $- 2 y$ and $9 y$ .

$- 6 + 7 y - y^{3} - 24 + 12 y = 0$

Simplify by adding terms.

Subtract $24$ from $- 6$ .

$7 y - y^{3} - 30 + 12 y = 0$

Add $7 y$ and $12 y$ .

$19 y - y^{3} - 30 = 0$

Step 2

Factor the left side of the equation.

Factor $- 1$ out of $19 y - y^{3} - 30$ .

Reorder $19 y$ and $- y^{3}$ .

$- y^{3} + 19 y - 30 = 0$

Factor $- 1$ out of $- y^{3}$ .

$- (y^{3}) + 19 y - 30 = 0$

Factor $- 1$ out of $19 y$ .

$- (y^{3}) - (- 19 y) - 30 = 0$

Rewrite $- 30$ as $- 1 (30)$ .

$- (y^{3}) - (- 19 y) - 1 \cdot 30 = 0$

Factor $- 1$ out of $- (y^{3}) - (- 19 y)$ .

$- (y^{3} - 19 y) - 1 \cdot 30 = 0$

Factor $- 1$ out of $- (y^{3} - 19 y) - 1 (30)$ .

$- (y^{3} - 19 y + 30) = 0$

Factor $y^{3} - 19 y + 30$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6 q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$2^{3} - 19 \cdot 2 + 30$

Raise $2$ to the power of $3$ .

$8 - 19 \cdot 2 + 30$

Multiply $- 19$ by $2$ .

$8 - 38 + 30$

Subtract $38$ from $8$ .

$- 30 + 30$

Add $- 30$ and $30$ .

$0$

Since $2$ is a known root, divide the polynomial by $y - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{y^{3} - 19 y + 30}{y - 2}$

Divide $y^{3} - 19 y + 30$ by $y - 2$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$y$

$2$

$y^{3}$

$0 y^{2}$

$19 y$

$30$

Divide the highest order term in the dividend $y^{3}$ by the highest order term in divisor $y$ .

					$y^{2}$
	$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$

Multiply the new quotient term by the divisor.

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			+	$y^{3}$	-	$2 y^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $y^{3} - 2 y^{2}$

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$

Divide the highest order term in the dividend $2 y^{2}$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					+	$2 y^{2}$	-	$4 y$

The expression needs to be subtracted from the dividend, so change all the signs in $2 y^{2} - 4 y$

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$
							-	$15 y$

Pull the next terms from the original dividend down into the current dividend.

				$y^{2}$	+	$2 y$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$
							-	$15 y$	+	$30$

Divide the highest order term in the dividend $- 15 y$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$2 y$	-	$15$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$
							-	$15 y$	+	$30$

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$2 y$	-	$15$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$
							-	$15 y$	+	$30$
							-	$15 y$	+	$30$

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 y + 30$

				$y^{2}$	+	$2 y$	-	$15$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$
							-	$15 y$	+	$30$
							+	$15 y$	-	$30$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y^{2}$	+	$2 y$	-	$15$
$y$	-	$2$		$y^{3}$	+	$0 y^{2}$	-	$19 y$	+	$30$
			-	$y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$19 y$
					-	$2 y^{2}$	+	$4 y$
							-	$15 y$	+	$30$
							+	$15 y$	-	$30$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$y^{2} + 2 y - 15$

Write $y^{3} - 19 y + 30$ as a set of factors.

$- ((y - 2) (y^{2} + 2 y - 15)) = 0$

Factor.

Factor $y^{2} + 2 y - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

$- ((y - 2) ((y - 3) (y + 5))) = 0$

Remove unnecessary parentheses.

$- ((y - 2) (y - 3) (y + 5)) = 0$

Remove unnecessary parentheses.

$- (y - 2) (y - 3) (y + 5) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 2 = 0$

$y - 3 = 0$

$y + 5 = 0$

Step 4

Set $y - 2$ equal to $0$ and solve for $y$ .

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Add $2$ to both sides of the equation.

$y = 2$

Step 5

Set $y - 3$ equal to $0$ and solve for $y$ .

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Add $3$ to both sides of the equation.

$y = 3$

Step 6

Set $y + 5$ equal to $0$ and solve for $y$ .

Set $y + 5$ equal to $0$ .

$y + 5 = 0$

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 7

The final solution is all the values that make $- (y - 2) (y - 3) (y + 5) = 0$ true.

$y = 2, 3, - 5$

Solve for y (1-y)((2-y)(-3-y))-24+12y=0

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