Find the x and y Intercepts f(x)=-8x+x^4-6x^3+5x^2

$f (x) = - 8 x + x^{4} - 6 x^{3} + 5 x^{2}$

Find the x-intercepts.

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

$0 = - 8 x + x^{4} - 6 x^{3} + 5 x^{2}$

Solve the equation.

Rewrite the equation as $- 8 x + x^{4} - 6 x^{3} + 5 x^{2} = 0$ .

$- 8 x + x^{4} - 6 x^{3} + 5 x^{2} = 0$

Reorder the polynomial.

$x^{4} - 6 x^{3} + 5 x^{2} - 8 x$

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.

$- 8 \cdot 0 + {(0)}^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$

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

Simplify each term.

Multiply $- 8$ by $0$ .

$0 + {(0)}^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$

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

$0 + 0 - 6 {(0)}^{3} + 5 {(0)}^{2}$

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

$0 + 0 - 6 \cdot 0 + 5 {(0)}^{2}$

Multiply $- 6$ by $0$ .

$0 + 0 + 0 + 5 {(0)}^{2}$

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

$0 + 0 + 0 + 5 \cdot 0$

Multiply $5$ by $0$ .

$0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0 + 0$

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.

$\frac{- 8 x + x^{4} - 6 x^{3} + 5 x^{2}}{x + 0}$

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.

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0				\xa0

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

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\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 $(- 6)$ .

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\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.

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$			\xa0
	$1$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(5)$ .

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$	$0$		\xa0
	$1$	$- 6$

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

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$	$0$		\xa0
	$1$	$- 6$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 8)$ .

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$1$	$- 6$	$5$

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

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$1$	$- 6$	$5$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$1$	$- 6$	$5$	$- 8$

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

$0$	$1$	$- 6$	$5$	$- 8$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$1$	$- 6$	$5$	$- 8$	$0$

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

$1 x^{3} + - 6 x^{2} + (5) x - 8$

Simplify the quotient polynomial.

$x^{3} - 6 x^{2} + 5 x - 8$

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

$x \approx 5.34453736$

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

$(x + 0) (x - 5.34453736)$

These are the roots (zeros) of the polynomial $x^{4} - 6 x^{3} + 5 x^{2} - 8 x$ .

$x = 0, 5.34453736$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (5.34453736, 0)$

Find the y-intercepts.

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

$y = - 8 \cdot 0 + {(0)}^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$

Solve the equation.

Remove parentheses.

$y = - 8 \cdot 0 + 0^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$

Remove parentheses.

$y = - 8 \cdot 0 + 0^{4} - 6 \cdot 0^{3} + 5 {(0)}^{2}$

Remove parentheses.

$y = - 8 \cdot 0 + 0^{4} - 6 \cdot 0^{3} + 5 \cdot 0^{2}$

Remove parentheses.

$y = - 8 \cdot 0 + {(0)}^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$

Simplify $- 8 \cdot 0 + {(0)}^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$ .

Simplify each term.

Multiply $- 8$ by $0$ .

$y = 0 + {(0)}^{4} - 6 {(0)}^{3} + 5 {(0)}^{2}$

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

$y = 0 + 0 - 6 {(0)}^{3} + 5 {(0)}^{2}$

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

$y = 0 + 0 - 6 \cdot 0 + 5 {(0)}^{2}$

Multiply $- 6$ by $0$ .

$y = 0 + 0 + 0 + 5 {(0)}^{2}$

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

$y = 0 + 0 + 0 + 5 \cdot 0$

Multiply $5$ by $0$ .

$y = 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 0$

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): $(0, 0), (5.34453736, 0)$

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

Find the x and y Intercepts f(x)=-8x+x^4-6x^3+5x^2

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