Factoring polynomial

Question

Can't figure out how to solve this problem:

using the factors (x+5) and (x+2), find the remaining factor(s) of f(x) = x³ + 6x² + 3x - 10 and write the polynomials in fully factored form.

Answer 1

-5 | 1 6 3 -10

-5 -5 10

-2 1 1 -2 0

-2 2

1 -1 0

x - 1 is the third factor

Answer 2

Problem:

using the factors (x+5) and (x+2), find the remaining factor(s) of f(x) = x³ + 6x² + 3x - 10 and write the polynomials in fully factored form.

Given:

(x+5) and (x+2) are factors of f(x) = x³ + 6x² + 3x - 10.

First approach:

Try to factor f(x) and see if it is simple to figure out....hmmm it's not

Note: f(x) has and x³ term which means there are 3 factors for this equation you only need to find one

Second Approach:

Use Synthetic Division to simplify the equation with one of the factors.

Note: If you need to look up Synthetic Devision technique in your Algebra 2 book or search youtube.

Use the Factor (x+5) with synthetic division

x³ +6x² +3x -10

| | | |

v v v v

________________________

-5 | 1 6 3 -10

__-5__ -5__+10______

1 1 -2 0

=> x² + x -2

We know the other factor is (x+2), we we can factor

the above equation

(x+2) (x-1)

Answer: (x+5)(x+2)(x-1)

to double check your answer, graph it, or multiply the factors and see if you get the original equation

Answer 3

f(x) = x³ + 6x² + 3x - 10 = (x+5)(x+2)(Ax+b) {a quadratic of x³ has three factors)

Note that (5)(2)(b) = -19 So, b=-1

Also, (x)(x)(Ax) = x³ so. A=1

The factors are: (x+5)(x+2)(x-1)

You could have expanded (x+5)(x+2) and divided the result into x³+6x²+3x-10 using synthetic division or long division, but remembering F-O-I-L is quite helpful