How Do You Use The Factor Theorem

Factor Theorem

Theorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then

(i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).

Proof: By the Remainder Theorem,

p(x) = (x – a) q(x) + p(a).

(i) If p(a) = 0, then p(x) = (x – a) q(x),

which shows that x – a is a factor of p(x).

(ii) Since x – a is a factor of p(x),

p(x) = (x – a) g(x) for same polynomial g(x).

In this case, p(a) = (a – a) g(a) = 0.

To use factor theorem

• Step 1: (x + a) is factor of a polynomial p(x) if p(–a) = 0.
• Step 2: (ax – b) is a factor of a polynomial p(x) if p(b/a) = 0
• Step 3: ax + b is a factor of a polynomial p(x) if p(–b/a) = 0.
• Step 4: (x – a) (x – b) is a factor of a polynomial p(x) if p(a) = 0 and p(b) = 0.

Factor Theorem Example Problems With Solutions

Example 1:    Examine whether x + 2 is a factor of x³ + 3x² + 5x + 6 and of 2x + 4.

Solution:    The zero of x + 2 is –2.

Let p(x) = x³ + 3x² + 5x + 6 and s(x) = 2x + 4

Then,    p(–2) = (–2)³ + 3(–2)² + 5(–2) + 6

= –8 + 12 – 10 + 6

= 0

So, by the Factor Theorem, x + 2 is a factor of x³ + 3x² + 5x + 6.

Again,   s(–2) = 2(–2) + 4 = 0

So, x + 2 is a factor of 2x + 4.

Example 2:    Use the factor theorem to determine whether x – 1 is a factor of

(a) x³ + 8x² – 7x – 2

(b) 2x³ + 5x² – 7

(c) 8x⁴ + 12x³ – 18x + 14

Solution:

Example 3:    Factorize each of the following expression, given that x³ + 13 x² + 32 x + 20. (x+2) is a factor.

Solution:

Example 4:    Factorize x3 – 23 x² + 142 x – 120

Solution:

Example 5:    Show that (x – 3) is a factor of the polynomial x³ – 3x² + 4x – 12

Solution:

Example 6:    Show that (x – 1) is a factor of x¹⁰ – 1 and also of x¹¹ – 1.

Solution:

Example 7:    Show that x + 1 and 2x – 3 are factors of 2x³ – 9x² + x + 12.

Solution:

Example 8:    Find the value of k, if x + 3 is a factor of 3x² + kx + 6.

Solution:   

Example 9:    If ax³ + bx² + x – 6 has x + 2 as a factor and leaves a remainder 4 when divided by (x – 2), find the values of a and b.

Solution:

Example 10:    If both x – 2 and x – 1/2 are factors of px² + 5x + r, show that p = r.

Solution:

Example 11:    If x² – 1 is a factor of ax⁴ + bx³ + cx² + dx + e, show that a + c + e = b + d = 0.

Solution:

Example 12:    Using factor theorem, show that a – b, b – c and c – a are the factors of a(b² – c²) + b(c² – a²) + c(a² – b²).

Solution: