# HOW TO FIND A CUBIC POLYNOMIAL WHOSE ZEROS ARE GIVEN

All cubic are continuous smooth curves. Every cubic polynomial can be categorized into one of four types.

Type 1 :

Three distinct real roots :

p(x) = a (x - p) (x - q) (x - r)

Type 2 :

Two real zeroes, one is repeated :

p(x) = a (x - p)2 (x - q)

Type 3 :

One real zero repeated three times.

p(x) = a (x - p)3

Type 4 :

One real and two imaginary zeroes.

p(x) = (x - p)(ax2 + bx + c)

Δ = b- 4ac

Find all cubic polynomials with zeros of :

Problem 1 :

±2, 3

Solution :

The zeroes of required polynomials are -2, 2 and 3. That is,

x = -2, x = 2 and x = 3

Converting them as factors.

(x + 2) (x - 2) and (x - 3)

By multiplying the factor.

P (x) = a[(x2 - 2x + 2x - 4) (x - 3)]

= a[(x2 - 4) (x - 3)]

= a[x3 - 3x2 - 4x + 12]

= a[x2(x - 3) - 4(x - 3)]

P(x) = a[(x2 - 4) (x - 3)] where a ≠ 0

Hence the required polynomial is a(x2 - 4) (x - 3).

Problem 2 :

-2, ± i

Solution :

The zeroes of required polynomials are -2, 2 and 3.

That is,

x = -2, x = i and x = -i

Converting them as factors.

(x + 2) (x - i) and (x + i)

By multiplying the factor.

P(x) = a[(x2 - xi + 2x - 2i) (x + i)]

= a[x3 + ix2 - x2i - xi2 + 2x2 + 2xi  - 2ix - 2i2]

= a[x3 + 2x2 + x + 2]

= a[x2(x + 2) + 1(x + 2)]

= a[(x2 + 1) (x + 2)] where a ≠ 0

Hence the required polynomial is a(x2 - 4) (x - 3).

Problem 3 :

3, -1 ± i

Solution :

The zeroes of required polynomials are 3, -1 + i and -1 - i. That is,

x = 3, x = -1 + i and x = -1 - i

Finding the quadratic factor with -1 + i and -1 - i

Sum of roots = -1 + i - 1 - i ==>  -2

Product of roots = (-1 + i) (-1 - i)

= 12 - i2 ==> 2

Quadratic factor :

y = x2 - 2x + 2

Linear factor :

(x - 3)

= a(x2 + 2x + 2) where a ≠ 0

Hence the required polynomial is a(x - 3) (x2 + 2x + 2).

Problem 4 :

-1, -2 ± √2

Solution :

The zeroes of required polynomials are -1, -2 + √2  and -2 - √2. That is,

x = -1, x = -2 + √2 and x = -2 - √2

Sum of the roots = (-2 + √2 - 2 - √2)

= -4

Products of the roots = (-2 + √2) (-2 - √2)

= 4 + 2√2 - 2√2 - 2

= 2

P(x) = a[x2 - (sum of the roots)x + (products of the roots)]

= a(x2 + 4x + 2) where a ≠ 0

Hence the required polynomial is a(x + 1) (x2 + 4x + 2).

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