Factor 5x^4+x^3+3x^2-7

$5 x^{4} + x^{3} + 3 x^{2} - 7$

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.

$p = \pm 1, \pm 7 q = \pm 1, \pm 5$

Step 2

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

$\pm 1, \pm 0.2, \pm 7, \pm 1.4$

Step 3

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

Substitute $- 1$ into the polynomial.

$5 {(- 1)}^{4} + {(- 1)}^{3} + 3 {(- 1)}^{2} - 7$

Raise $- 1$ to the power of $4$ .

$5 \cdot 1 + {(- 1)}^{3} + 3 {(- 1)}^{2} - 7$

Multiply $5$ by $1$ .

$5 + {(- 1)}^{3} + 3 {(- 1)}^{2} - 7$

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

$5 - 1 + 3 {(- 1)}^{2} - 7$

Subtract $1$ from $5$ .

$4 + 3 {(- 1)}^{2} - 7$

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

$4 + 3 \cdot 1 - 7$

Multiply $3$ by $1$ .

$4 + 3 - 7$

Add $4$ and $3$ .

$7 - 7$

Subtract $7$ from $7$ .

$0$

Step 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.

$\frac{5 x^{4} + x^{3} + 3 x^{2} - 7}{x + 1}$

Step 5

Divide $5 x^{4} + x^{3} + 3 x^{2} - 7$ by $x + 1$ .

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

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $x$ .

$5 x^{3}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

Multiply the new quotient term by the divisor.

$5 x^{3}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + 5 x^{3}$

$5 x^{3}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

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

$5 x^{3}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

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

$5 x^{3}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

$5 x^{3}$

$4 x^{2}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

Multiply the new quotient term by the divisor.

$5 x^{3}$

$4 x^{2}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} - 4 x^{2}$

$5 x^{3}$

$4 x^{2}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

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

$5 x^{3}$

$4 x^{2}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

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

$5 x^{3}$

$4 x^{2}$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

$5 x^{3}$

$4 x^{2}$

$7 x$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$5 x^{3}$

$4 x^{2}$

$7 x$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 7 x$

$5 x^{3}$

$4 x^{2}$

$7 x$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

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

$5 x^{3}$

$4 x^{2}$

$7 x$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

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

$5 x^{3}$

$4 x^{2}$

$7 x$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

$7$

Divide the highest order term in the dividend $- 7 x$ by the highest order term in divisor $x$ .

$5 x^{3}$

$4 x^{2}$

$7 x$

$7$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

$7$

Multiply the new quotient term by the divisor.

$5 x^{3}$

$4 x^{2}$

$7 x$

$7$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

$7$

$7 x$

$7$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x - 7$

$5 x^{3}$

$4 x^{2}$

$7 x$

$7$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

$7$

$7 x$

$7$

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

$5 x^{3}$

$4 x^{2}$

$7 x$

$7$

$x$

$1$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$0 x$

$7$

$5 x^{4}$

$5 x^{3}$

$4 x^{3}$

$3 x^{2}$

$4 x^{3}$

$4 x^{2}$

$7 x^{2}$

$0 x$

$7 x^{2}$

$7 x$

$7$

$7 x$

$7$

$0$

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

$5 x^{3} - 4 x^{2} + 7 x - 7$

Step 6

Write $5 x^{4} + x^{3} + 3 x^{2} - 7$ as a set of factors.

$(x + 1) (5 x^{3} - 4 x^{2} + 7 x - 7)$

Factor 5x^4+x^3+3x^2-7

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