Find the x and y Intercepts h(x)=x^4-3x^2+x

$h (x) = x^{4} - 3 x^{2} + x$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Solve the equation.

Rewrite the equation as $x^{4} - 3 x^{2} + x = 0$ .

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

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.

$p = \pm 0$

$q = \pm 1$

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

$0$

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(0)}^{4} - 3 {(0)}^{2} + 0$

Simplify the expression. In this case, the expression is equal to $0$ so $x = 0$ is a root of the polynomial.

Remove parentheses.

${(0)}^{4} - 3 {(0)}^{2} + 0$

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$0 - 3 {(0)}^{2} + 0$

Raising $0$ to any positive power yields $0$ .

$0 - 3 \cdot 0 + 0$

Multiply $- 3$ by $0$ .

$0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 3 x^{2} + x}{x + 0}$

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$			\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 3)$ .

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$	$0$		\xa0
	$1$	$0$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$	$0$		\xa0
	$1$	$0$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(1)$ .

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$1$	$0$	$- 3$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$1$	$0$	$- 3$	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$1$	$0$	$- 3$	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$0$	$- 3$	$1$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$1$	$0$	$- 3$	$1$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

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

Simplify the quotient polynomial.

$x^{3} - 3 x + 1$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.87938524, 0.34729635, 1.53208888$

The polynomial can be written as a set of linear factors.

$(x + 0) (x + 1.87938524) (x - 0.34729635) (x - 1.53208888)$

These are the roots (zeros) of the polynomial $x^{4} - 3 x^{2} + x$ .

$x = 0, - 1.87938524, 0.34729635, 1.53208888$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 1.87938524, 0), (0.34729635, 0), (1.53208888, 0)$

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = {(0)}^{4} - 3 {(0)}^{2} + 0$

Solve the equation.

Remove parentheses.

$y = {(0)}^{4} - 3 {(0)}^{2} + 0$

Remove parentheses.

$y = 0^{4} - 3 {(0)}^{2} + 0$

Remove parentheses.

$y = 0^{4} - 3 \cdot 0^{2} + 0$

Remove parentheses.

$y = {(0)}^{4} - 3 {(0)}^{2} + 0$

Simplify ${(0)}^{4} - 3 {(0)}^{2} + 0$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

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

Raising $0$ to any positive power yields $0$ .

$y = 0 - 3 \cdot 0 + 0$

Multiply $- 3$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

List the intersections.

x-intercept(s): $(0, 0), (- 1.87938524, 0), (0.34729635, 0), (1.53208888, 0)$

y-intercept(s): $(0, 0)$

Find the x and y Intercepts h(x)=x^4-3x^2+x

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