Given the function P(x)=(x-3)^2(x-7), find its y-i

Questions: Given the function P(x)=(x-3)^2(x-7), find its y-intercept is and its x-intercepts are x1= and x2= with x1<x2 When x → ∞, y → ∞ (Input + or - for the answer) When x → -∞, y → ∞ (Input + or - for the answer)

Transcript text: Given the function $P(x)=(x-3)^{2}(x-7)$, find its $y$-intercept is $\square$ its $x$-intercepts are $x_{1}=$ $\square$ and $x_{2}=$ $\square$ with $x_{1}

Solution

Solution Steps

Step 1: Finding the y-intercept

To find the y-intercept of the polynomial, we substitute $x = 0$ into the polynomial function. Thus, the y-intercept is $P(0) = -63$, which means the point is $(0, -63)$.

Step 2: Finding the x-intercepts

For higher-degree polynomials, finding the x-intercepts analytically may not be feasible. Numerical methods or specialized algorithms are required.

Step 3: Behavior as x approaches infinity or negative infinity

Since the degree of the polynomial is odd and the leading coefficient is positive, the polynomial approaches infinity as $x$ approaches infinity and approaches negative infinity as $x$ approaches negative infinity.