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)(ax^{2} + bx + c)
Δ = b^{2 }- 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[(x^{2} - 2x + 2x - 4) (x - 3)]
= a[(x^{2} - 4) (x - 3)]
= a[x^{3} - 3x^{2} - 4x + 12]
= a[x^{2}(x - 3) - 4(x - 3)]
P(x) = a[(x^{2} - 4) (x - 3)] where a ≠ 0
Hence the required polynomial is a(x^{2} - 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[(x^{2} - xi + 2x - 2i) (x + i)]
= a[x^{3} + ix^{2} - x^{2}i - xi^{2} + 2x^{2} + 2xi - 2ix - 2i^{2}]
= a[x^{3} + 2x^{2} + x + 2]
= a[x^{2}(x + 2) + 1(x + 2)]
= a[(x^{2} + 1) (x + 2)] where a ≠ 0
Hence the required polynomial is a(x^{2} - 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)
= 1^{2} - i^{2 }==> 2
Quadratic factor :
y = x^{2 } - 2x + 2
Linear factor :
(x - 3)
= a(x^{2} + 2x + 2) where a ≠ 0
Hence the required polynomial is a(x - 3) (x^{2} + 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[x^{2} - (sum of the roots)x + (products of the roots)]
= a(x^{2} + 4x + 2) where a ≠ 0
Hence the required polynomial is a(x + 1) (x^{2} + 4x + 2).
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM