Algebra long division

Question

Given that x=2 is zero of P(x)=x^3+x^2-16x+20 , find all other zeros of P(x) by using long division

Answer 1

x^3 +x^2 -16x +20 = P(x)

x=2 P(2) = 0

2^3 +2^2 -16(2) +20 = 8+4-32 +20 = 0

use either synthetic division or long division to find the quadratic polynomial

as the quotient when you divide P(x) by x-2

x^2 +3x -10 =0

factor

(x+5)(x-2) = 0

set each factor =0

x=2 and -5

there are two zeros, 2 and -5.

2 is a zero which repeats. That means if you graph P(x)

the curve will be tangent to the x axis at x=2 and intersect the x axis at x=-5

you could solve this problem by just graphing P(x), such as with Desmos graphing calculator and see where the curve touches the x axis, those are the zeros. the curve cuts the x axis at x=-5 and a U shaped part of the curve will be tangent at a local minimum of zero at x=2

It's worth graphing just to check your answer

or do the division as above.

sythetic division is more popular than long division.

but either works

2 | 1 1 -16 20

2 6 -20

1 3 -10 0

last row gives the coefficients of the quadratic quotient

x^2 +3x -10

1 times 2 = 2

1 +2 = 3

3 times 2 = 6

-16 +6 = -10

-10 times 2 = -20

20 + (-20) = 0

if you can't factor the quadratic, use the quadratic formula

x = -3/2 +/-(1/2)sqr(9+40)

x = (-3 +/-7)/2 = 4/2 or -10/2

x = 2 or -5

for long division x=2 and x-2 =0

use x-2 as the divisor, P(x) as the dividend

x^2 +3x -10 is the quotient

x-2 | x^3 +x^2 -16x +20

x^3-2x^2

3x^2 -16x

3x^2 -6x

-10x +20

-10x +20

it divides evenly with no remainder

it's just long regular long division in arithmetic,

except you have a letter variable instead of just all constant numbers