Question

1. choose the correct roots for each polynomial equation.

$x^3 + 2x^2 - 23x - 60 = (x + 3)(x + 4)(x - 5)$
$\circ$ $x = -3, -4, 5$
$\circ$ $x = 3, 4, 5$
$\circ$ $x = -3, -4, -5$
$\circ$ $x = 3, -4, 5$

1. choose the best answer.

the points where the graph of the polynomial crosses the $x$-axis are called ____ number roots.
$\circ$ factors
$\circ$ real
$\circ$ synthetic
$\circ$ constant

b. divide using synthetic division.
$\underline{-4\\,\big|\\,1\\;\\;7\\;\\;2\\;\\;-40}$
$\circ$ $x^2 + 3x - 10$
$\circ$ $-x^2 - 3x + 10$
$\circ$ $x + 3x - 10x$
$\circ$ $x^2 = 3x^2 - 10x$

Explanation:

Step1: Find roots of polynomial

Set each factor to 0:
$x+3=0 \implies x=-3$
$x+4=0 \implies x=-4$
$x-5=0 \implies x=5$

Step2: Identify x-axis crossing points

Points where a polynomial crosses the x-axis are its real roots, as these correspond to real x-values where the polynomial equals 0.

Step3: Perform synthetic division

Bring down the leading coefficient 1.
Multiply 1 by -4: $1 \times (-4) = -4$; add to 7: $7 + (-4) = 3$
Multiply 3 by -4: $3 \times (-4) = -12$; add to 2: $2 + (-12) = -10$
Multiply -10 by -4: $-10 \times (-4) = 40$; add to -40: $-40 + 40 = 0$
The resulting coefficients are 1, 3, -10, so the quotient is $x^2 + 3x - 10$.

Answer:

1. $x = -3, -4, 5$
2. real
3. $x^2 + 3x - 10$