Question

Each function, list the x-int er (x)=x^(5)+4x^(4)+4x^(3)

Answer 1

The x-intercept for the function
`\( f(x) = x^5 + 4x^4 + 4x^3 \)` is ( x = 0 ).

Step-by-step explanation:

To find the x-intercept of a function, we set ( f(x) ) equal to zero and solve for ( x ). In this case, we have the equation
`\( x^5 + 4x^4 + 4x^3 = 0 \)`. Factoring out the common factor of
`\( x^3 \), we get \( x^3(x^2 + 4x + 4) = 0 \).` Setting each factor equal to zero gives us two possible solutions: ( x = 0 ) and
`\( x^2 + 4x + 4 = 0 \).`

However, when we solve for ( x ) in the quadratic equation
`\( x^2 + 4x + 4 = 0 \)`, we find that it can be factored into
`\( (x + 2)^2 = 0 \)`. This yields a repeated root of ( x = -2 ). Therefore, the only distinct x-intercept is ( x = 0 ).

In mathematical terms, the function
`\( f(x) = x^5 + 4x^4 + 4x^3 \)` has a triple root at ( x = 0 ) and a double root at ( x = -2 ). The repeated roots indicate that the graph touches the x-axis at these points without crossing it. Therefore, the final x-intercept for this function is ( x = 0 ).