1. Explain the Factor Theorem.
Now that you understand how to use the Remainder Theorem to find the remainder of polynomials without actual division, the next theorem to look at in this article is called the Factor Theorem.
What is a Factor Theorem?
Consider a polynomial f (x) of degree k ≥ 1. If the term ‘a’ is any real number, then we can state that;
(x – k) is a factor of f (x), if f (k) = 0.
Proof of the Factor Theorem
Given that f (x) is a polynomial being divided by (x – k), if f (k) = 0 then,
f(x) = (x – k) q(x) + f(k)
f(x) = (x – k) q(x) + 0
f(x) = (x – k) q(x)
Hence, (x – k) is a factor of the polynomial f (x).
Hence, the Factor Theorem is a special case of Remainder Theorem, which states that a polynomial f (x) has a factor x- k, if and only if, k is a root i.e., f (k) = 0.
2. Use the Factor Theorem to show that (x + 2) and (x - 4) are factors of f(x).
f(x) = 8x^4 -14x^3 -71x^2 - 10x + 24
If x + 2 is a factor, then (setting this factor equal to zero and solving) x = -2 is a root.
To do the required verification, I need to check that, when I use synthetic division on f (x), with x = -2, I get a zero remainder:
.........(8x^3-30x^2-11x+12
(x+2)|8x^4 -14x^3 -71x^2 - 10x + 24
.........8x^4+16x^3...................subtract
..................-30x^3..........bring the next term down
..................-30x^3-71x^2
..................-30x^3-60x^2...............subtract
............................-11x^2........bring the next term down
............................-11x^2- 10x
............................-11x^2- 22x..............subtract
...........................................12x.......bring the next term down
...........................................12x+24
...........................................12x+24............subtract
......................................................0-> reminder
The remainder is zero, so the Factor Theorem says that:
x +2 is a factor of f(x) = 8x^4 -14x^3 -71x^2 - 10x + 24
do same for (x-4)
.........(8x^3 +18x^2+x- 6
(x-4)|8x^4 -14x^3 -71x^2 - 10x + 24
........8x^4 -32x^3 ...........subtract
................. 18x^3 ......bring the next term down
................. 18x^3 -71x^2
................. 18x^3 -72x^2...........subtract
..................................x^2....bring the next term down
..................................x^2- 10x
..................................x^2- 4x .........subtract
.......................................- 6x......bring the next term down
.......................................- 6x+ 24
.......................................- 6x+ 24........subtract
..................................................0-> reminder
The remainder is zero, so the Factor Theorem says that:
x -4 is a factor of f(x) =8x^4 -14x^3 -71x^2 - 10x + 24
