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.
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