Form A Polynomial With The Given Zeros
Let zeros of a quadratic polynomial be α and β.
x = β, x = β
x – α = 0, x – β = 0
The obviously the quadratic polynomial is
(x – α) (x – β)
i.e., x2 – (α + β) x + αβ
x2 – (Sum of the zeros)x + Product of the zeros
Form A Polynomial With The Given Zeros Example Problems With Solutions
Example 1: Form the quadratic polynomial whose zeros are 4 and 6.
Sol. Sum of the zeros = 4 + 6 = 10
Product of the zeros = 4 × 6 = 24
Hence the polynomial formed
= x2 – (sum of zeros) x + Product of zeros
= x2 – 10x + 24
Example 2: Form the quadratic polynomial whose zeros are –3, 5.
Sol. Here, zeros are – 3 and 5.
Sum of the zeros = – 3 + 5 = 2
Product of the zeros = (–3) × 5 = – 15
Hence the polynomial formed
= x2 – (sum of zeros) x + Product of zeros
= x2 – 2x – 15
Example 3: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively (frac { 1 }{ 2 }), – 1
Sol. Let the polynomial be ax2 + bx + c and its zeros be α and β.
(i) Here, α + β = (frac { 1 }{ 4 }) and α.β = – 1
Thus the polynomial formed
= x2 – (Sum of zeros) x + Product of zeros
(=^{text{2}}}-left( frac{1}{4} right)text{x}-1=^{text{2}}}-frac{text{x}}{text{4}}-1)
The other polynomial are (text{k}left( ^{text{2}}}text{-}frac{text{x}}{text{4}}text{-1} right))
If k = 4, then the polynomial is 4x2 – x – 4.
Example 4: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively (sqrt { 2 }), (frac { 1 }{ 3 })
Sol. Here, α + β =(sqrt { 2 }), αβ = (frac { 1 }{ 3 })
Thus the polynomial formed
= x2 – (Sum of zeroes) x + Product of zeroes
= x2 – (sqrt { 2 }) x + (frac { 1 }{ 3 })
Other polynomial are (text{k}left( ^{text{2}}}text{-}frac{text{x}}{text{3}}text{-1} right))
If k = 3, then the polynomial is
3x2 – (3sqrt { 2 }x) + 1
Example 5: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively 0, √5
Sol. Here, α + β = 0, αβ = √5
Thus the polynomial formed
= x2 – (Sum of zeroes) x + Product of zeroes
= x2 – (0) x + √5 = x2 + √5
Example 6: Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time, and product of its zeroes as 2, – 7 and –14, respectively.
Sol. Let the cubic polynomial be ax3 + bx2 + cx + d
⇒ x3 + (frac { b }{ a })x2 + (frac { c }{ a })x + (frac { d }{ a }) …(1)
and its zeroes are α, β and γ then
α + β + γ = 2 = (frac { -b }{ a })
αβ + βγ + γα = – 7 = (frac { c }{ a })
αβγ = – 14 = (frac { -d }{ a })
Putting the values of (frac { b }{ a }), (frac { c }{ a }), and (frac { d }{ a }) in (1), we get
x3 + (–2) x2 + (–7)x + 14
⇒ x3 – 2x2 – 7x + 14
Example 7: Find the cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and product of its zeroes as 0, –7 and –6 respectively.
Sol. Let the cubic polynomial be ax3 + bx2 + cx + d
⇒ x3 + (frac { b }{ a })x2 + (frac { c }{ a })x + (frac { d }{ a }) …(1)
and its zeroes are α, β and γ then
α + β + γ = 0 = (frac { -b }{ a })
αβ + βγ + γα = – 7 = (frac { c }{ a })
αβγ = – 6 = (frac { -d }{ a })
Putting the values of (frac { b }{ a }), (frac { c }{ a }), and (frac { d }{ a }) in (1), we get
x3 – (0) x2 + (–7)x + (–6)
⇒ x3 – 7x + 6
Example 8: If α and β are the zeroes of the polynomials ax2 + bx + c then form the polynomial whose zeroes are (frac { 1 }{ alpha } quad andquad frac { 1 }{ beta } )
Since α and β are the zeroes of ax2 + bx + c
So α + β = (frac { -b }{ a }) , α β = (frac { c }{ a })
Sum of the zeroes = (frac { 1 }{ alpha } +frac { 1 }{ beta } =frac { alpha +beta }{ alpha beta } )
(=frac{frac{-b}{c}}{frac{c}{a}}=frac{-b}{c})
Product of the zeroes
(=frac{1}{alpha }.frac{1}{beta }=frac{1}{frac{c}{a}}=frac{a}{c})
But required polynomial is
x2 – (sum of zeroes) x + Product of zeroes
(Rightarrow ^{2}}-left( frac{-b}{c} right)text{x}+left( frac{a}{c} right))
(Rightarrow ^{2}}+frac{b}{c}text{x}+frac{a}{c})
(Rightarrow cleft( ^{2}}+frac{b}{c}text{x}+frac{a}{c} right))
⇒ cx2 + bx + a
