Question

1. write a polynomial equation for a graph that passes through the point (-1,60) and has three x-intercepts: (-4,0), (1,0), and (3,0). step 1: start by writing an equation with those intercepts using the zero product property. p(x) = a(x + 4)(x - 1)(x - 3) step 2: find the stretch factor by substituting the point: (60) = a(-1 + 4)(-1 - 1)(-1 - 3) 60 = a(3)(-2)(-4) 60 = a(24) a = \frac{5}{2}

Explanation:

Step1: Form intercept-based polynomial

$p(x) = a(x+4)(x-1)(x-3)$

Step2: Substitute point $(-1,60)$

$60 = a((-1)+4)((-1)-1)((-1)-3)$

Step3: Simplify right-hand side

$60 = a(3)(-2)(-4)$
$60 = 24a$

Step4: Solve for stretch factor $a$

$a = \frac{60}{24} = \frac{5}{2}$

Step5: Substitute $a$ into polynomial

$p(x) = \frac{5}{2}(x+4)(x-1)(x-3)$

Step6: Expand the polynomial

First multiply $(x+4)(x-1)=x^2+3x-4$
Then multiply $(x^2+3x-4)(x-3)=x^3-3x^2+3x^2-9x-4x+12=x^3-13x+12$
Finally: $p(x) = \frac{5}{2}x^3 - \frac{65}{2}x + 30$

Answer:

$p(x) = \frac{5}{2}(x+4)(x-1)(x-3)$ or expanded form $p(x) = \frac{5}{2}x^3 - \frac{65}{2}x + 30$