Write as a Set of Linear Factors f(x)=2x^3+x^2-54x+72

$f (x) = 2 x^{3} + x^{2} - 54 x + 72$

Step 1

Factor $2 x^{3} + x^{2} - 54 x + 72$ 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.

$f (x) = p = \pm 1, \pm 72, \pm 2, \pm 36, \pm 3, \pm 24, \pm 4, \pm 18, \pm 6, \pm 12, \pm 8, \pm 9$

$q = \pm 1, \pm 2$

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

$f (x) = \pm 1, \pm 0.5, \pm 72, \pm 36, \pm 2, \pm 18, \pm 3, \pm 1.5, \pm 24, \pm 12, \pm 4, \pm 9, \pm 6, \pm 8, \pm 4.5$

Substitute $1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $1.5$ is a root of the polynomial.

Substitute $1.5$ into the polynomial.

$f (x) = 2 \cdot {1.5}^{3} + {1.5}^{2} - 54 \cdot 1.5 + 72$

Raise $1.5$ to the power of $3$ .

$f (x) = 2 \cdot 3.375 + {1.5}^{2} - 54 \cdot 1.5 + 72$

Multiply $2$ by $3.375$ .

$f (x) = 6.75 + {1.5}^{2} - 54 \cdot 1.5 + 72$

Raise $1.5$ to the power of $2$ .

$f (x) = 6.75 + 2.25 - 54 \cdot 1.5 + 72$

Add $6.75$ and $2.25$ .

$f (x) = 9 - 54 \cdot 1.5 + 72$

Multiply $- 54$ by $1.5$ .

$f (x) = 9 - 81 + 72$

Subtract $81$ from $9$ .

$f (x) = - 72 + 72$

Add $- 72$ and $72$ .

$f (x) = 0$

Since $1.5$ is a known root, divide the polynomial by $2 x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$f (x) = \frac{2 x^{3} + x^{2} - 54 x + 72}{2 x - 3}$

Divide $2 x^{3} + x^{2} - 54 x + 72$ by $2 x - 3$ .

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 $2 x^{3}$ by the highest order term in divisor $2 x$ .

$f (x) =$

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 $2 x^{3} - 3 x^{2}$

$f (x) =$

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 $4 x^{2}$ by the highest order term in divisor $2 x$ .

$f (x) =$

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 $4 x^{2} - 6 x$

$f (x) =$

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 $- 48 x$ by the highest order term in divisor $2 x$ .

$f (x) =$

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 $- 48 x + 72$

$f (x) =$

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.

$f (x) = x^{2} + 2 x - 24$

Write $2 x^{3} + x^{2} - 54 x + 72$ as a set of factors.

$f (x) = (2 x - 3) (x^{2} + 2 x - 24)$

Step 2

Factor $x^{2} + 2 x - 24$ using the AC method.

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 24$ and whose sum is $2$ .

$f (x) = - 4, 6$

Write the factored form using these integers.

$f (x) = (2 x - 3) ((x - 4) (x + 6))$

Remove unnecessary parentheses.

$f (x) = (2 x - 3) (x - 4) (x + 6)$

Write as a Set of Linear Factors f(x)=2x^3+x^2-54x+72

Related Questions

Support LCMGCF.com by disabling your adblocker.