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

Solution:

Determine the x-intercepts

Step 1: Identify the x-intercepts

To locate the x-intercepts, let $y=0$ and find the corresponding $x$ values.

Step 1.1: Set up the equation

$0=(x^2-5)(x+6)$

Step 1.2: Proceed to solve the equation

Step 1.2.1: Express the equation as

$(x^2-5)(x+6)=0$

Step 1.2.2: Apply the zero product property

$x^2-5=0$ or $x+6=0$

Step 1.2.3: Solve for $x$ when $x^2-5=0$

Step 1.2.3.1: Set the first factor equal to zero

$x^2-5=0$

Step 1.2.3.2: Isolate $x^2$

Step 1.2.3.2.1: Add $5$ to both sides

$x^2=5$

Step 1.2.3.2.2: Extract the square root

$x=\pm \sqrt{5}$

Step 1.2.3.2.3: Consider both positive and negative roots

Step 1.2.3.2.3.1: Positive square root gives

$x=\sqrt{5}$

Step 1.2.3.2.3.2: Negative square root gives

$x=-\sqrt{5}$

Step 1.2.3.2.3.3: Combine both solutions

$x=\sqrt{5},-\sqrt{5}$

Step 1.2.4: Solve for $x$ when $x+6=0$

Step 1.2.4.1: Set the second factor equal to zero

$x+6=0$

Step 1.2.4.2: Isolate $x$

$x=-6$

Step 1.2.5: Compile all $x$ values that satisfy the equation

$x=\sqrt{5},-\sqrt{5},-6$

Step 1.3: Express x-intercepts as coordinate points

x-intercepts: $(\sqrt{5},0),(-\sqrt{5},0),(-6,0)$

Determine the y-intercepts

Step 2: Identify the y-intercepts

To find the y-intercepts, let $x=0$ and solve for $y$.

Step 2.1: Set up the equation

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

Step 2.2: Solve the equation

Step 2.2.1: Expand the equation

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

Step 2.2.2: Simplify the equation

$y=(0-5)(0+6)$

Step 2.2.3: Further simplify

$y=(-5)(6)$

Step 2.2.4: Calculate the product

$y=-30$

Step 2.3: Express y-intercepts as coordinate points

y-intercept: $(0,-30)$

Compile the list of intercepts

x-intercepts: $(\sqrt{5},0),(-\sqrt{5},0),(-6,0)$ y-intercept: $(0,-30)$

Knowledge Notes:

To solve for the x-intercepts of a function, we set $y=0$ and solve the resulting equation for $x$. The x-intercepts are the points where the graph of the function crosses the x-axis. Similarly, to find the y-intercepts, we set $x=0$ and solve for $y$. The y-intercepts are the points where the graph crosses the y-axis.

In this problem, we have a quadratic function multiplied by a linear function. The zero product property states that if a product of factors equals zero, at least one of the factors must be zero. We use this property to find the x-intercepts by setting each factor equal to zero and solving for $x$.

For the y-intercept, we substitute $x=0$ into the function and simplify to find the value of $y$ when the graph crosses the y-axis.

The solutions to the equations provide the intercepts, which are expressed as coordinate points on the Cartesian plane.