Algebra Examples

$f (x) = - x^{4} + x^{3} + 21 x^{2} - 41 x + 20$

Step 1

Set $- x^{4} + x^{3} + 21 x^{2} - 41 x + 20$ equal to $0$ .

$- x^{4} + x^{3} + 21 x^{2} - 41 x + 20 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$- x^{4} + x^{3} + 21 x^{2} - 41 x + 20 = 0$

Step 2.1.2

Factor $- x^{3}$ out of $- x^{4} + x^{3}$ .

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Step 2.1.2.1

Factor $- x^{3}$ out of $- x^{4}$ .

$- x^{3} x + x^{3} + 21 x^{2} - 41 x + 20 = 0$

Step 2.1.2.2

Factor $- x^{3}$ out of $x^{3}$ .

$- x^{3} x - x^{3} \cdot - 1 + 21 x^{2} - 41 x + 20 = 0$

Step 2.1.2.3

Factor $- x^{3}$ out of $- x^{3} (x) - x^{3} \cdot - 1$ .

$- x^{3} (x - 1) + 21 x^{2} - 41 x + 20 = 0$

Step 2.1.3

Factor by grouping.

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Step 2.1.3.1

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 $a \cdot c = 21 \cdot 20 = 420$ and whose sum is $b = - 41$ .

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Step 2.1.3.1.1

Factor $- 41$ out of $- 41 x$ .

$- x^{3} (x - 1) + 21 x^{2} - 41 x + 20 = 0$

Step 2.1.3.1.2

Rewrite $- 41$ as $- 20$ plus $- 21$

$- x^{3} (x - 1) + 21 x^{2} + (- 20 - 21) x + 20 = 0$

Step 2.1.3.1.3

Apply the distributive property.

$- x^{3} (x - 1) + 21 x^{2} - 20 x - 21 x + 20 = 0$

Step 2.1.3.2

Factor out the greatest common factor from each group.

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Step 2.1.3.2.1

Group the first two terms and the last two terms.

$- x^{3} (x - 1) + 21 x^{2} - 20 x - 21 x + 20 = 0$

Step 2.1.3.2.2

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

$- x^{3} (x - 1) + x (21 x - 20) - (21 x - 20) = 0$

Step 2.1.3.3

Factor the polynomial by factoring out the greatest common factor, $21 x - 20$ .

$- x^{3} (x - 1) + (21 x - 20) (x - 1) = 0$

Step 2.1.4

Factor $x - 1$ out of $- x^{3} (x - 1) + (21 x - 20) (x - 1)$ .

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Step 2.1.4.1

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

$(x - 1) (- x^{3}) + (21 x - 20) (x - 1) = 0$

Step 2.1.4.2

Factor $x - 1$ out of $(21 x - 20) (x - 1)$ .

$(x - 1) (- x^{3}) + (x - 1) (21 x - 20) = 0$

Step 2.1.4.3

Factor $x - 1$ out of $(x - 1) (- x^{3}) + (x - 1) (21 x - 20)$ .

$(x - 1) (- x^{3} + 21 x - 20) = 0$

Step 2.1.5

Factor.

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Step 2.1.5.1

Rewrite $- x^{3} + 21 x - 20$ in a factored form.

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Step 2.1.5.1.1

Factor $- x^{3} + 21 x - 20$ using the rational roots test.

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Step 2.1.5.1.1.1

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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1$

Step 2.1.5.1.1.2

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 2.1.5.1.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.1.5.1.1.3.1

Substitute $1$ into the polynomial.

$- 1^{3} + 21 \cdot 1 - 20$

Step 2.1.5.1.1.3.2

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

$- 1 \cdot 1 + 21 \cdot 1 - 20$

Step 2.1.5.1.1.3.3

Multiply $- 1$ by $1$ .

$- 1 + 21 \cdot 1 - 20$

Step 2.1.5.1.1.3.4

Multiply $21$ by $1$ .

$- 1 + 21 - 20$

Step 2.1.5.1.1.3.5

Add $- 1$ and $21$ .

$20 - 20$

Step 2.1.5.1.1.3.6

Subtract $20$ from $20$ .

$0$

Step 2.1.5.1.1.4

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{3} + 21 x - 20}{x - 1}$

Step 2.1.5.1.1.5

Divide $- x^{3} + 21 x - 20$ by $x - 1$ .

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Step 2.1.5.1.1.5.1

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

$x$

$1$

$x^{3}$

$0 x^{2}$

$21 x$

$20$

Step 2.1.5.1.1.5.2

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

				-	$x^{2}$
	$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$

Step 2.1.5.1.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			-	$x^{3}$	+	$x^{2}$

Step 2.1.5.1.1.5.4

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$

Step 2.1.5.1.1.5.5

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$

Step 2.1.5.1.1.5.6

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$

Step 2.1.5.1.1.5.7

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$

Step 2.1.5.1.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					-	$x^{2}$	+	$x$

Step 2.1.5.1.1.5.9

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$

Step 2.1.5.1.1.5.10

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$
							+	$20 x$

Step 2.1.5.1.1.5.11

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

			-	$x^{2}$	-	$x$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$
							+	$20 x$	-	$20$

Step 2.1.5.1.1.5.12

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	-	$x$	+	$20$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$
							+	$20 x$	-	$20$

Step 2.1.5.1.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$x$	+	$20$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$
							+	$20 x$	-	$20$
							+	$20 x$	-	$20$

Step 2.1.5.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $20 x - 20$

			-	$x^{2}$	-	$x$	+	$20$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$
							+	$20 x$	-	$20$
							-	$20 x$	+	$20$

Step 2.1.5.1.1.5.15

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

			-	$x^{2}$	-	$x$	+	$20$
$x$	-	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	-	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$21 x$
					+	$x^{2}$	-	$x$
							+	$20 x$	-	$20$
							-	$20 x$	+	$20$
										$0$

Step 2.1.5.1.1.5.16

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

$- x^{2} - x + 20$

Step 2.1.5.1.1.6

Write $- x^{3} + 21 x - 20$ as a set of factors.

$(x - 1) ((x - 1) (- x^{2} - x + 20)) = 0$

Step 2.1.5.1.2

Factor by grouping.

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Step 2.1.5.1.2.1

Factor by grouping.

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Step 2.1.5.1.2.1.1

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 $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = - 1$ .

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Step 2.1.5.1.2.1.1.1

Factor $- 1$ out of $- x$ .

$(x - 1) ((x - 1) (- x^{2} - (x) + 20)) = 0$

Step 2.1.5.1.2.1.1.2

Rewrite $- 1$ as $4$ plus $- 5$

$(x - 1) ((x - 1) (- x^{2} + (4 - 5) x + 20)) = 0$

Step 2.1.5.1.2.1.1.3

Apply the distributive property.

$(x - 1) ((x - 1) (- x^{2} + 4 x - 5 x + 20)) = 0$

Step 2.1.5.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.5.1.2.1.2.1

Group the first two terms and the last two terms.

$(x - 1) ((x - 1) ((- x^{2} + 4 x) - 5 x + 20)) = 0$

Step 2.1.5.1.2.1.2.2

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

$(x - 1) ((x - 1) (x (- x + 4) + 5 (- x + 4))) = 0$

Step 2.1.5.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 4$ .

$(x - 1) ((x - 1) ((- x + 4) (x + 5))) = 0$

Step 2.1.5.1.2.2

Remove unnecessary parentheses.

$(x - 1) ((x - 1) (- x + 4) (x + 5)) = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

$(x - 1) (x - 1) (- x + 4) (x + 5) = 0$

Step 2.1.6

Combine exponents.

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Step 2.1.6.1

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

$(x - 1) (x - 1) (- x + 4) (x + 5) = 0$

Step 2.1.6.2

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

$(x - 1) (x - 1) (- x + 4) (x + 5) = 0$

Step 2.1.6.3

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

${(x - 1)}^{1 + 1} (- x + 4) (x + 5) = 0$

Step 2.1.6.4

Add $1$ and $1$ .

${(x - 1)}^{2} (- x + 4) (x + 5) = 0$

Step 2.2

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

${(x - 1)}^{2} = 0$

$- x + 4 = 0$

$x + 5 = 0$

Step 2.3

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set ${(x - 1)}^{2}$ equal to $0$ .

${(x - 1)}^{2} = 0$

Step 2.3.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

Set $- x + 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $- x + 4$ equal to $0$ .

$- x + 4 = 0$

Step 2.4.2

Solve $- x + 4 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 2.4.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 2.4.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 2.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 2.5

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

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Step 2.5.1

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

$x + 5 = 0$

Step 2.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.6

The final solution is all the values that make ${(x - 1)}^{2} (- x + 4) (x + 5) = 0$ true.

$x = 1, 4, - 5$

Step 3