Write as a Set of Linear Factors f(x)=x^3-17x^2+55x+25

$f (x) = x^{3} - 17 x^{2} + 55 x + 25$

Step 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.

$f (x) = p = \pm 1, \pm 25, \pm 5$

$q = \pm 1$

Step 2

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

$f (x) = \pm 1, \pm 25, \pm 5$

Step 3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

Substitute $5$ into the polynomial.

$f (x) = 5^{3} - 17 \cdot 5^{2} + 55 \cdot 5 + 25$

Raise $5$ to the power of $3$ .

$f (x) = 125 - 17 \cdot 5^{2} + 55 \cdot 5 + 25$

Raise $5$ to the power of $2$ .

$f (x) = 125 - 17 \cdot 25 + 55 \cdot 5 + 25$

Multiply $- 17$ by $25$ .

$f (x) = 125 - 425 + 55 \cdot 5 + 25$

Subtract $425$ from $125$ .

$f (x) = - 300 + 55 \cdot 5 + 25$

Multiply $55$ by $5$ .

$f (x) = - 300 + 275 + 25$

Add $- 300$ and $275$ .

$f (x) = - 25 + 25$

Add $- 25$ and $25$ .

$f (x) = 0$

Step 4

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$f (x) = \frac{x^{3} - 17 x^{2} + 55 x + 25}{x - 5}$

Step 5

Divide $x^{3} - 17 x^{2} + 55 x + 25$ by $x - 5$ .

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 5 x^{2}$

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} + 60 x$

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x + 25$

$f (x) =$

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

$f (x) =$

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

$f (x) = x^{2} - 12 x - 5$

Step 6

Write $x^{3} - 17 x^{2} + 55 x + 25$ as a set of factors.

$f (x) = (x - 5) (x^{2} - 12 x - 5)$

Write as a Set of Linear Factors f(x)=x^3-17x^2+55x+25

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