1. ## Factor Theorem, help?

In an explanation text about “The Factor Theorem” the following example is given:

Example: A polynomial is given by f(x) = 2x3 + 13x2 + 13x – 10.
c) Factorise f(x) completely.
Solution:
2x3 + 13x2 +13x – 10 = (x + 2)(ax2 +bx + c).
This is an identity and so the values of a, b and c may be found by comparing coefficients.

Comparing the x3 term gives: a = 2.
Comparing the x2 term gives: 13 = 2a + b → b = 9.
Comparing the constant term gives: c = -5.
If follows that: f(x) = (x + 2)(2x2 + 9x – 5).
Factorising the quadratic part in the usual way gives:
f(x) = (x + 2)(2x – 1)(x + 5).
My questions are:

a) where does the expression “2a + b” come from in the line “comparing the x2 term gives” and
b) how is the c = -5 calculated?

I can see that 2x3 + 13x2 becomes 2a + b if a is = x3 and b = x2 but why? What are the rules being exercised?

NB 2x3 is 2x cubed, 13x2 is 13x squared, see attached file
Many thanks

Steve Gould

2. Originally Posted by stephen_gould
In an explanation text about “The Factor Theorem” the following example is given:

Example: A polynomial is given by f(x) = 2x3 + 13x2 + 13x – 10.
c) Factorise f(x) completely.
Solution:
2x3 + 13x2 +13x – 10 = (x + 2)(ax2 +bx + c).
This is an identity and so the values of a, b and c may be found by comparing coefficients.

Comparing the x3 term gives: a = 2.
Comparing the x2 term gives: 13 = 2a + b → b = 9.
Comparing the constant term gives: c = -5.
If follows that: f(x) = (x + 2)(2x2 + 9x – 5).
Factorising the quadratic part in the usual way gives:
f(x) = (x + 2)(2x – 1)(x + 5).
My questions are:

a) where does the expression “2a + b” come from in the line “comparing the x2 term gives” and
b) how is the c = -5 calculated?
You have:

2x^3 + 13x^2 +13x – 10 = (x + 2)(ax^2 +bx + c).

expanding the right hand side:

2x^3 + 13x^2 +13x – 10 =ax^3+(2a+b)x^2+(2b+c)x+2c

Now look at the coefficient of x^2 on both sides of this identity these
must be equal so:

13=2a+b.

And now look at the constant terms on both sides, these also must be
equal so:

-10=2c,

so c=-5.

RonL

3. ## got it, thanks

Got it thanks.