Algebra Examples

$y^{3} - 27 = 9 y^{2} - 27 y$

Step 1

Solve for $y$ .

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Step 1.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 y^{2} - 27 y = y^{3} - 27$

Step 1.2

Subtract $y^{3}$ from both sides of the equation.

$9 y^{2} - 27 y - y^{3} = - 27$

Step 1.3

Add $27$ to both sides of the equation.

$9 y^{2} - 27 y - y^{3} + 27 = 0$

Step 1.4

Factor the left side of the equation.

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Step 1.4.1

Reorder terms.

$- y^{3} + 9 y^{2} - 27 y + 27 = 0$

Step 1.4.2

Factor $- y^{3} + 9 y^{2} - 27 y + 27$ using the rational roots test.

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Step 1.4.2.1

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.

$p = \pm 1, \pm 27, \pm 3, \pm 9$

$q = \pm 1$

Step 1.4.2.2

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

$\pm 1, \pm 27, \pm 3, \pm 9$

Step 1.4.2.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 1.4.2.3.1

Substitute $3$ into the polynomial.

$- 3^{3} + 9 \cdot 3^{2} - 27 \cdot 3 + 27$

Step 1.4.2.3.2

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

$- 1 \cdot 27 + 9 \cdot 3^{2} - 27 \cdot 3 + 27$

Step 1.4.2.3.3

Multiply $- 1$ by $27$ .

$- 27 + 9 \cdot 3^{2} - 27 \cdot 3 + 27$

Step 1.4.2.3.4

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

$- 27 + 9 \cdot 9 - 27 \cdot 3 + 27$

Step 1.4.2.3.5

Multiply $9$ by $9$ .

$- 27 + 81 - 27 \cdot 3 + 27$

Step 1.4.2.3.6

Add $- 27$ and $81$ .

$54 - 27 \cdot 3 + 27$

Step 1.4.2.3.7

Multiply $- 27$ by $3$ .

$54 - 81 + 27$

Step 1.4.2.3.8

Subtract $81$ from $54$ .

$- 27 + 27$

Step 1.4.2.3.9

Add $- 27$ and $27$ .

$0$

Step 1.4.2.4

Since $3$ is a known root, divide the polynomial by $y - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- y^{3} + 9 y^{2} - 27 y + 27}{y - 3}$

Step 1.4.2.5

Divide $- y^{3} + 9 y^{2} - 27 y + 27$ by $y - 3$ .

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Step 1.4.2.5.1

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

$y$

$3$

$y^{3}$

$9 y^{2}$

$27 y$

$27$

Step 1.4.2.5.2

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

				-	$y^{2}$
	$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$

Step 1.4.2.5.3

Multiply the new quotient term by the divisor.

			-	$y^{2}$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			-	$y^{3}$	+	$3 y^{2}$

Step 1.4.2.5.4

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

			-	$y^{2}$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$

Step 1.4.2.5.5

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

			-	$y^{2}$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$

Step 1.4.2.5.6

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

			-	$y^{2}$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$

Step 1.4.2.5.7

Divide the highest order term in the dividend $6 y^{2}$ by the highest order term in divisor $y$ .

			-	$y^{2}$	+	$6 y$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$

Step 1.4.2.5.8

Multiply the new quotient term by the divisor.

			-	$y^{2}$	+	$6 y$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					+	$6 y^{2}$	-	$18 y$

Step 1.4.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 y^{2} - 18 y$

			-	$y^{2}$	+	$6 y$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$

Step 1.4.2.5.10

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

			-	$y^{2}$	+	$6 y$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$
							-	$9 y$

Step 1.4.2.5.11

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

			-	$y^{2}$	+	$6 y$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$
							-	$9 y$	+	$27$

Step 1.4.2.5.12

Divide the highest order term in the dividend $- 9 y$ by the highest order term in divisor $y$ .

			-	$y^{2}$	+	$6 y$	-	$9$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$
							-	$9 y$	+	$27$

Step 1.4.2.5.13

Multiply the new quotient term by the divisor.

			-	$y^{2}$	+	$6 y$	-	$9$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$
							-	$9 y$	+	$27$
							-	$9 y$	+	$27$

Step 1.4.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 y + 27$

			-	$y^{2}$	+	$6 y$	-	$9$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$
							-	$9 y$	+	$27$
							+	$9 y$	-	$27$

Step 1.4.2.5.15

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

			-	$y^{2}$	+	$6 y$	-	$9$
$y$	-	$3$	-	$y^{3}$	+	$9 y^{2}$	-	$27 y$	+	$27$
			+	$y^{3}$	-	$3 y^{2}$
					+	$6 y^{2}$	-	$27 y$
					-	$6 y^{2}$	+	$18 y$
							-	$9 y$	+	$27$
							+	$9 y$	-	$27$
										$0$

Step 1.4.2.5.16

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

$- y^{2} + 6 y - 9$

Step 1.4.2.6

Write $- y^{3} + 9 y^{2} - 27 y + 27$ as a set of factors.

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

Step 1.4.3

Factor.

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Step 1.4.3.1

Factor by grouping.

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Step 1.4.3.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 6$ .

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Step 1.4.3.1.1.1

Factor $6$ out of $6 y$ .

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

Step 1.4.3.1.1.2

Rewrite $6$ as $3$ plus $3$

$(y - 3) (- y^{2} + (3 + 3) y - 9) = 0$

Step 1.4.3.1.1.3

Apply the distributive property.

$(y - 3) (- y^{2} + 3 y + 3 y - 9) = 0$

Step 1.4.3.1.2

Factor out the greatest common factor from each group.

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Step 1.4.3.1.2.1

Group the first two terms and the last two terms.

$(y - 3) ((- y^{2} + 3 y) + 3 y - 9) = 0$

Step 1.4.3.1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.4.3.1.3

Factor the polynomial by factoring out the greatest common factor, $- y + 3$ .

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

Step 1.4.3.2

Remove unnecessary parentheses.

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

Step 1.4.4

Combine exponents.

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Step 1.4.4.1

Factor $- 1$ out of $y$ .

$(- 1 (- y) - 3) (- y + 3) (y - 3) = 0$

Step 1.4.4.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 1.4.4.3

Factor $- 1$ out of $- 1 (- y) - 1 (3)$ .

$- 1 (- y + 3) (- y + 3) (y - 3) = 0$

Step 1.4.4.4

Raise $- y + 3$ to the power of $1$ .

$- 1 ((- y + 3) (- y + 3)) (y - 3) = 0$

Step 1.4.4.5

Raise $- y + 3$ to the power of $1$ .

$- 1 ((- y + 3) (- y + 3)) (y - 3) = 0$

Step 1.4.4.6

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

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

Step 1.4.4.7

Add $1$ and $1$ .

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

Step 1.4.4.8

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

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

Step 1.4.4.9

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.4.4.10

Factor $- 1$ out of $- (y) - 1 (- 3)$ .

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

Step 1.4.4.11

Rewrite $- (y - 3)$ as $- 1 (y - 3)$ .

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

Step 1.4.4.12

Apply the product rule to $- 1 (y - 3)$ .

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

Step 1.4.4.13

Raise $- 1$ to the power of $2$ .

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

Step 1.4.4.14

Multiply ${(y - 3)}^{2}$ by $1$ .

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

Step 1.4.4.15

Raise $y - 3$ to the power of $1$ .

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

Step 1.4.4.16

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

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

Step 1.4.4.17

Add $1$ and $2$ .

$- 1 {(y - 3)}^{3} = 0$

Step 1.5

Divide each term in $- 1 {(y - 3)}^{3} = 0$ by $- 1$ and simplify.

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Step 1.5.1

Divide each term in $- 1 {(y - 3)}^{3} = 0$ by $- 1$ .

$\frac{- 1 {(y - 3)}^{3}}{- 1} = \frac{0}{- 1}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Dividing two negative values results in a positive value.

$\frac{{(y - 3)}^{3}}{1} = \frac{0}{- 1}$

Step 1.5.2.2

Divide ${(y - 3)}^{3}$ by $1$ .

${(y - 3)}^{3} = \frac{0}{- 1}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $- 1$ .

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

Step 1.6

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

$y - 3 = 0$

Step 1.7

Add $3$ to both sides of the equation.

$y = 3$

Step 2

Set $9 y^{2} - 27 y$ equal to $0$ .

$3 = 0$

Step 3

Since $3 \neq 0$ , there are no solutions.

No solution