COMPARING COEFFICIENTS OF POLYNOMIALS

Problem 1 :

2z – 3 is a factor of 2z3 – z2 + az – 3. Find a and all zeros of the cubic.

Solution :

Since 2z – 3 is a factor, 2z - 3 = 0

2z = 3

z = 3/2

Given, 2z3 – z2 + az – 3

z = 3/2 substitute the given equation.  

2(3/2)3 – (3/2)2 + a(3/2) – 3 = 0

2(27/8) - 9/4 + 3a/2 - 3 = 0

27/4 - 9/4 + 3a/2 - 12/4 = 0

6/4 + 3a/2 = 0

3/2 + 3a/2 = 0

3/2 = -3a/2

a = -1

So, the value of a is -1.

 (2z3 – z2 - z – 3) ÷ (2z - 3)

Using synthetic division.

syntdivi1

(2z2 + 2z + 2) = 0

2(z2 + z + 1) = 0

Dividing 2 on each sides.

z2 + z + 1 = 0

a = 1, b = 1, c = 1

Using quadratic formula.

b2 - 4ac = 12 - 4(1)(1)

= 12 - 4

= -3

z = -b ± b2 - 4ac2az = -1 ± -32(1)z = -1 ± i32

z = -1 ± i3

Problem 2 :

3z + 2 is a factor of 3z3 – z2 + [a + 1]z + a. Find a and all the zeros of the cubic.

Solution :

3z + 2 is a factor.

3z + 2 = 0

3z = -2

z = -2/3

Given, 3z3 – z2 + [a + 1]z + a

3z3 – z2 + az + z + a

z = -2/3 substitute the given equation.

3-233- -232+ a-23+ -23 + a = 03-827 - 49 - 2a3 - 23 + a = 0-89 -49 - 6a9 - 69 + 9a9 = 0-189 + 3a9 = 0-2 + a3 = 0-2 = -a3a = 6

So, the value of a is 6.

 (3z3 – z2 + [a + 1]z + a) ÷ (3z + 2)

3z3 - z2 + [6 + 1]z + 6 ÷ (3z + 2)

3z3 - z2 + 7z + 6 ÷ (3z + 2)

Using synthetic division.

syndivi2

3z2 - 3z + 9 = 0

3(z2 - z + 3) = 0

Equating each factor to 0, we get

z = 0

z(z2 - z + 3) = 0

a = 1, b = -1, c = 3

Using quadratic formula.

b2 - 4ac = (-1)2 - 4(1)(3)

= 1 - 12

= -11

z = -b ± b2 - 4ac2az = 1 ± -112(1)z = 1 ± i112

Problem 3 :

Both 2x + 1 and x – 2 are factors of P(x) = 2x4 + ax3 + bx2 – 12x – 8. Find a and b and all zeros of P(x).

Solution :

2x + 1 and x – 2 are factors

2x + 1 = 0

2x = -1

x = -1/2

x - 2 = 0

x = 2

P(x) = 2x4 + ax3 + bx2 – 12x – 8

x = -1/2 substitute the given equation.

P(-1/2) = 2(-1/2)4 + a(-1/2)3 + b(-1/2)2 – 12(-1/2) – 8 = 0

2(1/16) + a(-1/8) + b(1/4) + 6 - 8 = 0

1/8 - (a/8) + (b/4) - 2 = 0

-(a/8) + (b/4) + (1/8) - 2 = 0

-(a/8) + (b/4) = 15/8

-a + 2b = 15 -----(1)

x = 2 substitute the given equation.

P(x) = 2x4 + ax3 + bx2 – 12x – 8

2(2)4 + a(2)3 + b(2)2 – 12(2) – 8 = 0

 32 + 8a + 4b - 24 - 8 = 0

8a + 4b = 0

Dividing by 4, we get

2a + b = 0 ----(2)

2 (1) + (2) ==> 

-2a + 4b + 2a + b  = 30 + 0

5b = 30

b = 6

Applying the value of b in (1), we get

-a + 2(6) = 15

-a + 12 = 15

-a = 15 - 12

-a = 3

a = -3

P(x) = 2x4 - 3x3 + 6x2 – 12x – 8

equatingcoeffiofpolyq3

2x2 + 8 = 0

Dividing the equation by 2, we get

x2 + 4 = 0

x2 = -4

x = ±i2

The zeroes are x = ±i2, x = -1/2 x = 2

Problem 4 :

x + 3 and 2x – 1 are factors of 2x4 + ax3 + bx2 + ax + 3. Find a and b and hence determine all zeros of the quartic.

Solution :

x + 3 and 2x – 1 are factors.

x + 3 = 0

x = -3

2x - 1 = 0

2x = 1

x = 1/2

2x4 + ax3 + bx2 + ax + 3

x  = -3 substitute the given equation.

2(-3)4 + a(-3)3 + b(-3)2 + a(-3) + 3 = 0

2(81) - 27a + 9b - 3a + 3 = 0 

162 - 30a + 9b + 3 = 0

165 - 30a + 9b = 0

-10a + 9b = -165

-10a + 3b = -55    ---(1)

x = 1/2 substitute the given equation.

p12= 2124+a123+b122+a12+30= 2116+a18+b4+a2+30=18+a8+b4+a2+35a8 + b4 = -258Multiplying by 8 on both sides5a+2b8= -2585a + 2b = -25

5a + 2b = -25 ---(2)

(1) + 2(2)

-10a + 3b + 10a + 4b = -55 - 50

7b = 105

b = 15

Applying the value of n in (1), we get

5a + 30 = -25

5a = -55

a = -11

Problem 5 :

x3 + 3x2  - 9x + c has two identical linear factors. Prove that c is either 5 or -27 and factories the cubic into linear factors in each case.

Solution :

x3 + 3x2  - 9x + c

equatingcoeffiofpolyq4.png

If x = 1 is a solution, then x -1 will be a factor, then the value of c will be -5.

Since we cannot factories x2 + 4x + 5

equatingcoeffiofpolyq5.png

(x - 2) is a factor, then c will be -2.

(x3 + 3x2  - 9x + c) ÷ (x - 3)

syndivi5

(x - 3) is a factor, then the value of C is -27.

P(x) = x2 + 6x + 9

= (x + 3) and (x + 3)

= (x + 3)2

So, P(x) = (x  - 3) (x + 3)2.

Problem 6 :

3x3 + 4x2 – x + m has two identical linear factors. Find m and find the zeros of the polynomial in all possible cases.

Solution :

3x3 + 4x2 – x + m

equatingcoeffiofpolyq6.png

If (x - 1) is a factor, then the value of m will be -6. But the quadratic polynomial created by the left over will not have repeated factor.

If (x + 1) is a factor.

3x2 + x - 2  = 0

(3x - 2) and  (x + 1) = 0

3x - 2 = 0

3x = 2

x = 2/3

x + 1 = 0

x = -1

So, the repeated factor is (x + 1).

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