Factor Theorem

1. Let p(x) be a polynomial and a be any real number. If p(a) = 0, then (x–a) is a factor of p(x).

2. If (x–a) is a factor of p(x), then p(a) = 0

**EXERCISE 2.4 (NCERT Solution)**

**Q1: Determine which of the following polynomials has (x + 1) a factor :**

**(i) x**

^{3}+ x^{2}+ x + 1**(ii) x**

^{4}+ x^{3}+ x^{2}+ x + 1**(iii) x**

^{4}+ 3x^{3}+ 3x^{2}+ x + 1**(iv) x**

^{3}– x^{2}– (2 + √2)x + √2
Answer: If (x+1) is a factor of a polynomial p(x), then p(-1) must be equal to zero.

(i) Let p(x) = x

^{3}+ x^{2}+ x + 1
p(-1) = (-1)

^{3}+ (-1)^{2}+ (-1) + 1 = -1 + 1 -1 + 1 = 0
∴ (x-1) is a factor of polynomial x

^{3}+ x^{2}+ x + 1
(ii) Let p(x) = x

^{4}+ x^{3}+ x^{2}+ x + 1
p(-1) = (-1)

^{4}+ (-1)^{3}+ (-1)^{2}+ (-1) + 1 = 1 - 1 + 1 -1 + 1 = 1 ≠ 0
∴ (x-1) is not a factor of this polynomial.

(iii) Let p(x) = x

^{4}+ 3x^{3}+ 3x^{2}+ x + 1
p(-1) = (-1)

^{4}+ 3(-1)^{3}+ 3(-1)^{2}+ (-1) + 1
= 1 + 3(-1) + 3(1) -1 + 1 = 1 - 3 + 3 - 1 = 1 ≠ 0

∴ (x-1) is not a factor of this polynomial.

(iv) Let p(x) = x

^{3}– x^{2}– (2 + √2)x + √2
p(-1) = (-1)

^{3}– (-1)^{2}– (2 + √2)(-1) + √2
= -1 - 1 + 2 + √2 + √2 = 2√2 ≠ 0

∴ (x-1) is not a factor of this polynomial.

**Q2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:**

**(i) p(x) = 2x**

^{3}+ x^{2}– 2x – 1, g(x) = x + 1

**(ii) p(x) = x**

^{3}+ 3x^{2}+ 3x + 1, g(x) = x + 2

**(iii) p(x) = x**

^{3}– 4x^{2}+ x + 6, g(x) = x – 3

Answer:

(i) p(x) = 2x

^{3}+ x^{2}– 2x – 1, g(x) = x + 1
Zero of x+1 is -1. If g(x) is a factor of p(x) then p(-1) = 0

⇒ p(-1) = 2(-1)

^{3}+ (-1)^{2}– 2(-1) – 1 = -2 + 1 + 2 - 1 = 0
∴ g(x) is a factor of polynomial p(x).

(ii) p(x) = x

^{3}+ 3x^{2}+ 3x + 1, g(x) = x + 2
Zero of x + 2 is -2. If g(x) is a factor of p(x) then p(-2) = 0

⇒ p(-2) = (-2)

^{3}+ 3(-2)^{2}+ 3(-2) + 1 = -8 + 12 - 6 + 1 = -1 ≠ 0
∴ g(x) is not a factor of polynomial p(x).

(iii) p(x) = x

^{3}– 4x^{2}+ x + 6, g(x) = x – 3
Root of x-2 is 3. If g(x) is a factor of p(x) then p(3) = 0

⇒ p(3) = (3)

^{3}– 4(3)^{2}+ 3 + 6 = 27 - 4(9) + 3 + 6 = 27 - 36 + 3 + 6 = 0
∴ g(x) is a factor of polynomial p(x).

**Q3: Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:**

**(i) p(x) = x**

^{2}+ x + k

**(ii) p(x) = 2x**

^{2}+ kx + √2

**(iii) p(x) = kx**

^{2}– √2x + 1

**(iv) p(x) = kx**

^{2}– 3x + k
Answer:

According to factor theorem, if x -1 is a factor of p(x), then p(1) = 0.

(i) p(x) = x

^{2}+ x + k
⇒ p(1) = (1)

^{2}+ 1 + k = 0
⇒ 1 + 1 + k = 0

⇒

**k = -2**
(ii) p(x) = 2x

^{2}+ kx + √2
⇒ p(1) = 0

⇒ p(1) = 2(1)

^{2}+ k(1) + √2 = 0
⇒ 2 + k + √2 = 0

⇒ k = -2 - √2 =

**-(2 +√2)**
(iii) p(x) = kx

^{2}– √2x + 1
⇒ p(1) = 0

⇒ p(1) = k(1)

^{2}– √2(1) + 1 = 0
⇒ k - √2 + 1

⇒ k =

**√2 - 1**
(iv) p(x) = kx

^{2}– 3x + k
⇒ p(1) = 0

⇒ p(1) = k(1)

^{2}– 3(1) + k = 0
⇒ k - 3 + k = 0

⇒ 2k = 3

⇒

**k = 3/2****Q4. Factorise:**

**(i) 12x**

^{2}– 7x + 1**(ii) 2x**

^{2}+ 7x + 3**(iii) 6x**

^{2}+ 5x – 6**(iv) 3x**

^{2}– x – 4
Answer:

(i) 12x

^{2}– 7x + 1
Method I: By splitting method, let us find out two number p and q such that pq = 12
× 1

and p + q = -7

i.e. p = -4 and q = -3

⇒ = 12x

^{2}– 4x -3x + 1
= 4x(3x - 1) - 1(3x -1)

= (4x -1)(3x -1) ...(answer)

Method II: By factor theorem.

(ii) 2x

^{2}+ 7x + 3

Let us find out two number p and q such that pq = 2
× 3 = 6 and p + q = 7

i.e. p = 6 and q = 1

⇒ = 2x

^{2}+ 6x + x + 3
= 2x (x + 3) + 1(x + 3)

= (2x + 1)(x + 3) ... answer
(iii) 6x

^{2}+ 5x – 6
Let us find out two number p and q such that pq = 6
× -6 = -36 and p + q = 5

i.e. p = 9 and q = -4

∴ 6x

^{2}+ 5x – 6 = 6x^{2}+ 9x -4x – 6
= 3x(2x + 3) - 2(2x + 3)

= (2x + 3)(3x - 2) ... answer

(iv) 3x

^{2}– x – 4

Let us find out two number p and q such that pq = 3
× -4 = -12 and p + q = -1

i.e. p = 3 and q = -4

∴ 3x

^{2}– x – 4 = 3x^{2}+ 3x –4x – 4
= 3x(x + 1) -4(x+1)

= (x+1)(3x - 4) ... answer

**Q5. Factorise:**

**(i) x**

^{3}– 2x^{2}– x + 2**(ii) x**

^{3}– 3x^{2}– 9x – 5**(iii) x**

^{3}+ 13x^{2}+ 32x + 20**(iv) 2y**

^{3}+ y^{2}– 2y – 1
Answer:

(i) Let p(x) = x

^{3}– 2x^{2}– x + 2
Here constant term is 2. Possible factors of 2 are: ±1, ±2

By trial method, p(2) = (2)

^{3}– 2(2)^{2}– 2 + 2 = 8 - 8 - 2 + 2 = 0
∴ (x -2 ) is factor of p(x).

P(x) ÷ (x -2) =

x^{2}- 1

▁▁▁▁▁▁▁▁▁▁▁▁▁ x -2 ) x^{3}– 2x^{2}– x + 2 x^{3}- 2x^{3}

- + ▔▔▔▔▔▔▔▔ -x + 2 -x + 2 + - ▔▔▔▔▔ 0

Since, Dividend = Divisor × Quotient + Remainder

∴ x

^{3}– 2x^{2}– x + 2 = (x-2)(x^{2}– 1) + 0
= (x - 2)( x

^{2}-x + x– 1)
= (x -2) [x(x-1) + 1 (x-1)]

= (x - 2) (x-1)(x+1)

(ii) Let f(x) = x

^{3}– 3x^{2}– 9x – 5
Here constant is 5, Possible factors of 5 are ±1 and ±5

By trial method, p(5) = (5)

^{3}– 3(5)^{2}– 9(5) – 5 = 125 - 75 - 45 - 5 = 0
∴ (x - 5) is factor of polynomial f(x).

Let us find out quotient = f(x) ÷ (x -5)

x^{2 }+ 2x + 1 ▁▁▁▁▁▁▁▁▁▁ x - 5) x^{3}– 3x^{2}– 9x – 5 x^{3}- 5x^{2}

- +

▔▔▔▔▔▔

0 + 2x^{2}- 9x - 5

2x^{2}- 10x

- +

▔▔▔▔▔▔▔▔

x - 5

x - 5

▔▔▔▔▔

0

Since, Dividend = Divisor × Quotient + Remainder

∴ x

^{3}– 3x^{2}– 9x – 5 = (x - 5)(x^{2 }+ 2x + 1)
Applying splitting method,

= (x -5)( x

^{2 }+ x + x + 1)
= (x -5)[x(x+1) +1(x+1)]

= (x-5)(x+1)(x+1)

= (x-5)(x+1)^{2}

**Method II**:

x

^{3}– 3x^{2}– 9x – 5 = x^{3}– 5x^{2}+ 2x^{2}– 9x – 5
= x

^{2}(x – 5)^{}+ 2x^{2}– 10x + x – 5
= x

^{2}(x – 5) + 2x(x - 5) + 1 (x -5)
= (x - 5)( x

^{2}+ 2x + 1)
= (x -5)[x(x+1) +1(x+1)]

= (x-5)(x+1)(x+1)

= (x-5)(x+1)^{2}

(iii) Let p(x) = x

^{3}+ 13x^{2}+ 32x + 20
Here constant term is 20. Its factors are: ±1, ±2, ±4, ±5 etc.

By trial method, p(-2) = (-2)

^{3}+ 13(-2)^{2}+ 32(-2) + 20
= -8 + 52 -64 + 20 = 0

∴ x + 2 is a factor of polynomial p(x).

⇒ x

^{3}+ 13x^{2}+ 32x + 20 = x^{3}+ 2x^{2}+ 11x^{2}+ 22x + 10x + 20
= x

^{2}(x + 2) + 11x(x + 2) + 10(x+2)
Taking (x + 2) common, we get

= (x + 2)( x

^{2}+ 11x + 10)
(Note: you can use long division method also).

= (x + 2)(x

^{2}+ x + 10x + 10)
= (x+2)[x(x+1) + 10(x+1)]

= (x+2)(x+10)(x+1)

(iv) Let p(y) = 2y

^{3}+ y^{2}– 2y – 1
Here the constant term is 1. Possible factors are: ±1

By trial method, p(1) = 2(1)

So (y-1) is a factor of p(y)

Quotient = p(y) ÷ (y-1)

By trial method, p(1) = 2(1)

^{3}+ (1)^{2}– 2(1) – 1 = 2 + 1 - 2 - 1 = 0So (y-1) is a factor of p(y)

Quotient = p(y) ÷ (y-1)

2y^{2 }+ 3y + 1

_____________________ y -1 ) 2y^{3}+ y^{2}– 2y – 1 2y^{3}-2y^{2}- + ▔▔▔▔▔▔▔▔▔▔▔▔

3y^{2}- 2y - 1

3y^{2}- 3y

^{ - +}

▔▔▔▔▔▔▔▔▔▔▔▔

y - 1

y - 1- +▔▔▔▔▔▔▔▔▔▔▔▔0

Since, Dividend = Divisor × Quotient + Remainder

2y

= (y-1)(2y

= (y-1)(2y(y+1) + 1(y + 1))

= (y-1)(y+1)(2y + 1)

Applying remainder theorem, if (x - 1) is factor of p(x) then p(1) = 0.

⇒ p(x) = ax

⇒ p(1) = a(1)

⇒ a + b + c + d = 0

We can say that,

Answer: Adding the co-efficients i.e. 5 + (-4) + (-2) + 1 = 6 - 6 = 0

⇒ (x - 1) is a factor of the given polynomial.

2y

^{3}+ y^{2}– 2y – 1 = (y -1)(2y^{2}+ 3y + 1)= (y-1)(2y

^{2}+ 2y + y + 1)= (y-1)(2y(y+1) + 1(y + 1))

= (y-1)(y+1)(2y + 1)

**Criterion to check if (x - 1) is a factor of polynomial p(x) = ax**

^{3}+ bx^{2}+ cx + d, where a,b,c,d ∈ R, a≠0Applying remainder theorem, if (x - 1) is factor of p(x) then p(1) = 0.

⇒ p(x) = ax

^{3}+ bx

^{2}+ cx + d

⇒ p(1) = a(1)

^{3}+ b(1)

^{2}+ c(1) + d = 0

⇒ a + b + c + d = 0

We can say that,

❝(x - 1) is a factor of polynomial p(x), if the sum of all the coefficients of polynomial p(x) is zero.❞

(✏ By by applying similar logic, can you find criterion to check if (x + 1) is a factor?)

**Q6:Check if the polynomial 5x**^{4}- 4x^{3}- 2x + 1 has (x - 1) its factor.Answer: Adding the co-efficients i.e. 5 + (-4) + (-2) + 1 = 6 - 6 = 0

⇒ (x - 1) is a factor of the given polynomial.

can u suggest some other ways for question 5? i hav done another method but cant do same for question 5, 2nd

ReplyDeleteSee the update blog page. Method II is added for 5(ii). Hope it helps!

DeleteThis comment has been removed by the author.

Deletei want to understand factorise questions.

ReplyDeleteits simple just do with the help of long division and after wise factorise it

ReplyDeletewow! thanks a lot

ReplyDeleteThanks a lot, You really make it easy for me and may be for everyone...............

ReplyDeleteI'm in eighth but i studied the topic outta curiousity . however I was stuck on a question . This thing really helped !

ReplyDeletewhere can i find the other method of Q5

ReplyDeletethanku for making it easy

ReplyDelete