Chapter 2. Polynomials

Exercise 2.4

Q.1. Determine which of the following polynomials has (x+ 1) a factor:

(i) x³ +x² + x +1                                        (ii) x⁴ + x³ + x² + x +1

(iii) x⁴ + 3x³ + 3x² + x +1                             

Solution:

(i) If (x + 1) is a factor of p(x) = x³+ x²+ x + 1, then p (−1) must be zero, otherwise (x + 1) is not a factor of p(x).

P(x)  = x³ + x² + x + 1

P(−1)= (−1)³ + (−1)² + (−1) + 1

         = − 1 + 1 − 1 − 1

        = 0

∵ P(x) = 0 

Hence, x + 1 is a factor of this polynomial.

(ii) If (x + 1) is a factor of p(x) = x⁴+ x³+ x²+ x + 1, then p (−1) must be zero, Otherwise (x + 1) is not a factor of p(x).

P(x) = x⁴+ x³+ x²+ x + 1

P(−1) = (−1)⁴ + (−1)³ + (−1)² + (−1) + 1

          = 1 − 1 + 1 −1 + 1

          = 1

As P(x) ≠ 0, (− 1)

Therefore, x + 1 is not a factor of this polynomial.

(iii) If (x + 1) is a factor of polynomial p(x) = x⁴+ 3x³+ 3x²+ x + 1, then p (−1) must  be 0, otherwise (x + 1) is not a factor of this polynomial.

P(x) = x⁴+ 3x³+ 3x²+ x + 1

P(−1) = (−1)⁴+ 3(−1)³+ 3(−1)²+ (−1) + 1

          = 1 − 3 + 3 − 1 + 1

          = 1

As P(x) ≠ 0, (−1) 

Therefore (x+1) is not a factor of this polynomial .

Q.2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:

 (i)         P(x) = 2x³+ x²− 2x − 1, g(x) = x + 1

 (ii)        P(x) = x³ + 3x² + 3x + 1, g(x) = x + 2

 (iii)       P(x) = x³ − 4 x² +  x + 6, g(x) = x − 3 

Solution:

(i)   If g(x) = x + 1 is a factor of the given polynomial p(x), then p (−1) must be zero.

P (x) = 2x³ + x² − 2x − 1

P (−1) = 2(−1)³+ (−1)²− 2(−1) − 1

           = 2(−1) + 1 + 2 – 1   

           = 0

∵ P(x) = 0 

Hence, g(x) = x + 1 is a factor of the given polynomial.

(ii) If g(x) = x + 2 is a factor of the given polynomial p(x), then p (−2) must be 0.

P (x) = x³+3x²+ 3x + 1

P (−2) = (−2)³+ 3(−2)²+ 3(−2) + 1

           = − 8 + 12 − 6 + 1  

           = −1

As P(x) ≠ 0, 

Hence, g(x) = x + 2 is not a factor of the given polynomial.

(iii) If g(x) = x − 3 is a factor of the given polynomial p(x), then P(3) must be 0.

        P(x) = x³− 4 x²+ x + 6

        P(3) = (3)³ − 4(3)² + 3 + 6

                = 27 −36 + 9

                = 0

Hence, g(x) = x − 3 is a factor of the given polynomial. 

Solution:

: If x − 1 is a factor of polynomial p(x), then P(1) must be 0.

(i) P(x) = x²+ x + k

    P(1) = (1)²+ 1 + k

            = 1+1+ 

            = 2+k

        k  =−2 

(iv) P(x) = kx²-3x + k    

      P(1) = k(1)²-3(1) + k

             = k-3+k

        2k = 3

         K = 3/2

Question 4: Factorise: 

(i) 12x²− 7x + 1                           (ii) 2x²+ 7x + 3

(iii) 6x²+ 5x – 6                           (iv) 3x²− x − 4 

Solution:

(i) 12x²− 7x + 1 we can find two numbers,

Such that pq = 12 × 1 = 12 and p + q = −7.

They are p = −4 and q = −3

Here, 12x²− 7x + 1

      = 12x²− 4x − 3x + 1

      = 4x (3x − 1) − 1 (3x − 1)

      = (3x − 1) (4x − 1)

(ii) 2x²+ 7x + 3 we can find two numbers such that pq = 2 × 3= 6 and p + q = 7.

     They are p = 6 and q = 1.

     Here, 2x² + 7x + 3

    = 2x²+ 6x + x + 3

    = 2x (x + 3) + 1 (x + 3)

    = (x + 3) (2x+ 1) 

(iii) 6x²+ 5x − 6 we can find two numbers such that pq = −36 and p + q = 5.

     They are p = 9 and q = −4.

     Here, 6x²+ 5x – 6

    = 6x²+ 9x − 4x – 6

    = 3x (2x + 3) − 2 (2x + 3)

    = (2x + 3) (3x − 2) 

(iv)  3x²− x − 4 we can find two numbers,

such that pq = 3 × (−4) = −12 and p + q = −1.

They are p = −4 and q = 3

Here, 3x²− x − 4

       = 3x²− 4x + 3x – 4

       = x (3x − 4) + 1 (3x − 4)

       = (3x − 4) (x + 1)

Question 5. Factorize:

 (i) x³− 2x²− x + 2                 (ii) x³+ 3x²−9x − 5

(iii) x³+ 13x²+ 32x + 20         (iv) 2y³+ y²− 2y – 1 

Solution:

 (i) Let P(x) = x³− 2x²− x + 2 all the factor are there. These are ± 1, ± 2.

By trial method, P (1) = (1)³− 2(1)²− 1 + 2

         = 1 − 2 − 1+ 2

         = 0 Therefore, (x − 1) is factor of polynomial p(x)

         Let us find the quotient on dividing x³− 2x²− x + 2 by x − 1.

         By long division method

Now,

Dividend = Divisor × Quotient + remainder  

x³− 2x²− x + 2 = (x – 1) ( X²– x – 2) + 0

                       = (x – 1) (x²–2x+x–2)

                        = (x – 1) [x (x–2) + 1(x–2)]

                        = (x – 1) (x + 1) (x – 2)

(ii)   Let p(x) = x³ – 3x²−9x – 5 all the factor are there. These are ± 1, ± 2.

        By trial method, p (–1) = (–1)³– 3(1)²− 9(1) – 5

              = –1– 3–9–5     =0

        Therefore (x+1) is the factor of polynomial p(x).

       .   Let us find the quotient on dividing x³– 3x²−9x – 5 by x+1.

            By long division method

Now,

Dividend = Divisor × Quotient + remainder  

x³– 3x²−9x – 5 = (x +1) ( X²–4x – 5) + 0

                       =(x + 1) (x²–5x+x–5)

                        =(x + 1) [x (x–5) +1(x–5)]

                        =(x + 1) (x + 1) (x – 5) 

(iii)  Let p(x) = x³+ 13x²+ 32x + 20    all the factor are there.

These are ± 1, ± 2, ± 3, ± 4.

       By trial method, p (–1) = (–1)³+13(–1)²+ 32(–1) +20

                                           = –1+13–32+20   = 0    

       Therefore (x+1) is the factor of polynomial p(x).

        Let us find the quotient on dividing x³+ 13x²+ 32x + 20 by x+1

       By long division method

Now,

Dividend = Divisor × Quotient + remainder  

x³ +13x² + 32x + 20 = (x +1) ( x² + 12x + 20) + 0

                                 =(x + 1) (x²+10x+2x+20)

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

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

(iv) Let p(y) = 2y³+ y²− 2y – 1 all the factor are there. These are ± 1, ± 2.

       By trial method, p (1) =2(1)³ + (1)² – 2(1) – 1

                                        =2 + 1 – 2 – 1    =0

     Therefore (y–1) is the factor of polynomial p(y).

    Let us find the quotient on dividing 2y³+ y²− 2y – 1 by y–1.

    By long division method

Now,

Dividend = Divisor × Quotient + remainder

  2y³+ y²− 2y −1 =(y − 1) (2y²+3y + 1)

                           = (y − 1) (2y²+2y

                          = (y − 1) [2y (y+1) + 1 (y + 1)]

                          = (y − 1) (y + 1) (2y + 1)