Relationship Between Zeros And Coefficients Of A Polynomial
Consider quadratic polynomial
P(x) = 2x2 – 16x + 30.
Now, 2x2 – 16x + 30 = (2x – 6) (x – 3)
= 2 (x – 3) (x – 5)
The zeros of P(x) are 3 and 5.
Sum of the zeros = 3 + 5 = 8 = (frac { -left( -16 right) }{ 2 } ) = (text{-}left[ frac{text{coefficient of x}}{text{coefficient of }^{text{2}}}} right])
Product of the zeros = 3 × 5 = 15 = (frac { 30 }{ 2 }) = (left[ frac{text{constant term }}{text{coefficient of }^{text{2}}}} right])
So if ax2 + bx + c, a ≠ 0 is a quadratic polynomial and α, β are two zeros of polynomial then
(alpha +beta =-frac { b }{ a } )
(alpha beta =frac { c }{ a } )
In general, it can be proved that if α, β, γ are the zeros of a cubic polynomial ax3 + bx2 + cx + d, then
(alpha +beta +gamma =frac { -b }{ a } )
( alpha beta +beta gamma +gamma alpha =frac { c }{ a } )
( alpha beta gamma =frac { -d }{ a } )
Note: (frac { b }{ a } ), (frac { c }{ a }) and (frac { d }{ a } ) are meaningful because a ≠ 0.
Relationship Between Zeros And Coefficients Of A Polynomial Example Problems With Solutions
Example 1: Find the zeros of the quadratic polynomial 6x2 – 13x + 6 and verify the relation between the zeros and its coefficients.
Sol. We have, 6x2 – 13x + 6 = 6×2 – 4x – 9x + 6
= 2x (3x – 2) –3 (3x – 2)
= (3x – 2) (2x – 3)
So, the value of 6x2 – 13x + 6 is 0, when
(3x – 2) = 0 or (2x – 3) = 0 i.e.,
When x = (frac { 2 }{ 3 } ) or (frac { 3 }{ 2 } )
Therefore, the zeros of 6x2 – 13x + 6 are
(frac { 2 }{ 3 } ) and (frac { 3 }{ 2 } )
Sum of the zeros
= (frac { 2 }{ 3 } ) + (frac { 3 }{ 2 } ) = (frac { 13 }{ 6 } ) = (frac { left( -13 right) }{ 6 } ) = (text{-}left[ frac{text{coefficient of x}}{text{coefficient of }^{text{2}}}} right])
Product of the zeros
= (frac { 2 }{ 3 } ) × (frac { 3 }{ 2 } ) = (frac { 6 }{ 6 } ) = (left[ frac{text{constant term }}{text{coefficient of }^{text{2}}}} right])
Example 2: Find the zeros of the quadratic polynomial 4x² – 9 and verify the relation between the zeros and its coefficients.
Sol. We have,
4x2 – 9 = (2x)2 – 32 = (2x – 3) (2x + 3)
So, the value of 4x2 – 9 is 0, when
2x – 3 = 0 or 2x + 3 = 0
i.e., when x = (frac { 3 }{ 2 } ) or x = (frac { -3 }{ 2 } ).
Therefore, the zeros of 4x2 – 9 are (frac { 3 }{ 2 } ) & (frac { -3 }{ 2 } ).
Sum of the zeros
= (frac { 3 }{ 2 }) (-frac { 3 }{ 2 } ) = 0 = (-frac { left( 0 right) }{ 4 } ) = (text{-}left[ frac{text{coefficient of x}}{text{coefficient of }^{text{2}}}} right])
Product of the zeros
= (frac { 3 }{ 2 }) × (frac { -3 }{ 2 }) = (frac { -9 }{ 4 }) = (left[ frac{text{constant term }}{text{coefficient of }^{text{2}}}} right])
Example 3: Find the zeros of the quadratic polynomial 9x2 – 5 and verify the relation between the zeros and its coefficients.
Sol. We have,
9x2 – 5 = (3x)2 – (√5)2 = (3x – √5) (3x + √5)
So, the value of 9x2 – 5 is 0,
when 3x – √5 = 0 or 3x + √5 = 0
i.e., when x = (frac { sqrt { 5 } }{ 3 } ) or x = (frac { -sqrt { 5 } }{ 3 } ).
Sum of the zeros
= (frac { sqrt { 5 } }{ 3 } ) (-frac { sqrt { 5 } }{ 3 } ) = 0 = (-frac { left( 0 right) }{ 9 } ) = (text{-}left[ frac{text{coefficient of x}}{text{coefficient of }^{text{2}}}} right])
Product of the zeros
= (left( frac { sqrt { 5 } }{ 3 } right) ) × (left( frac { -sqrt { 5 } }{ 3 } right) ) = (frac { -5 }{ 9 } ) = (left[ frac{text{constant term }}{text{coefficient of }^{text{2}}}} right])
Example 4: If α and β are the zeros of ax2 + bx + c, a ≠ 0 then verify the relation between the zeros and its coefficients.
Sol. Since a and b are the zeros of polynomial ax2 + bx + c.
Therefore, (x – α), (x – β) are the factors of the polynomial ax2 + bx + c.
⇒ ax2 + bx + c = k (x – α) (x – β)
⇒ ax2 + bx + c = k {x2 – (α + β) x + αβ}
⇒ ax2 + bx + c = kx2 – k (α + β) x + kαβ …(1)
Comparing the coefficients of x2, x and constant terms of (1) on both sides, we get
a = k, b = – k (α + β) and c = kαβ
⇒ α + β = (frac { -b }{ k } ) and αβ = (frac { c }{ k } )
α + β = (frac { -b }{ a } ) and αβ = (frac { c }{ a } ) [∵ k = a]
Sum of the zeros = (frac { -b }{ a } ) = (frac{text{- coefficient of x}}{text{coefficient of }^{text{2}}}})
Product of the zeros = (frac { c }{ a } ) = (frac{text{constant term }}{text{coefficient of }^{text{2}}}})
Example 5: Prove relation between the zeros and the coefficient of the quadratic polynomial
ax2 + bx + c.
Sol. Let a and b be the zeros of the polynomial ax2 + bx + c
α = (frac{-b+sqrt-4ac}}{2a}) ….(1)
β = (frac{-b-sqrt-4ac}}{2a}) ….(2)
By adding (1) and (2), we get
α + β = (frac{-b+sqrt-4ac}}{2a}) + (frac{-b-sqrt-4ac}}{2a})
= (frac{ -2b }{ 2a } ) = (frac{ -b }{ a } ) = (frac{text{- coefficient of x}}{text{coefficient of }^{text{2}}}})
Hence, sum of the zeros of the polynomial
ax2 + bx + c is (frac{ -b }{ a } )
By multiplying (1) and (2), we get
αβ = (frac{-b+sqrt-4ac}}{2a}) × (frac{-b-sqrt-4ac}}{2a})
= (frac-+4ac}{4})
= (frac{4ac}{4}) = (frac{ c }{ a } )
= (frac{text{constant term }}{text{coefficient of }^{text{2}}}})
Hence, product of zeros = (frac{ c }{ a } )
Example 6: find the zeroes of the quadratic polynomial x2 – 2x – 8 and verify a relationship between zeroes and its coefficients.
Sol. x2 – 2x – 8 = x2 – 4x + 2x – 8
= x (x – 4) + 2 (x – 4) = (x – 4) (x + 2)
So, the value of x2 – 2x – 8 is zero when
x – 4 = 0 or x + 2 = 0 i.e., when x = 4 or x = – 2.
So, the zeroes of x2 – 2x – 8 are 4, – 2.
Sum of the zeroes
= 4 – 2 = 2 = (-frac { left( -2 right) }{ 1 } ) = (frac{text{- coefficient of x}}{text{coefficient of }^{text{2}}}})
Product of the zeroes
= 4 (–2) = –8 = (frac { -8 }{ 1 } ) = (frac{text{constant term }}{text{coefficient of }^{text{2}}}})
Example 7: Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients. 2x3 + x2 – 5x + 2 ; , 1, – 2
Sol. Here, the polynomial p(x) is 2x3 + x2 – 5x + 2
Value of the polynomial 2x3 + x2 – 5x + 2
when x = 1/2
= (2+-5left( frac{1}{2} right)+2) = (frac{1}{4}+frac{1}{4}-frac{5}{2}+2) = 0
So, 1/2 is a zero of p(x).
On putting x = 1 in the cubic polynomial
2x3 + x2 – 5x + 2
= 2(1)3 + (1)2 –¬ 5(1) + 2 = 2 + 1 – 5 + 2 = 0
On putting x = – 2 in the cubic polynomial
2x3 + x2 – 5x + 2
= 2(–2)3 + (–2)2 – 5 (–2) + 2
= – 16 + 4 + 10 + 2 = 0
Hence, (frac{ 1 }{ 2 } ), 1, – 2 are the zeroes of the given polynomial.
Sum of the zeroes of p(x)
= (frac{ 1 }{ 2 } ) + 1 – 2 = (frac{ -1 }{ 2 } ) = (frac{-text{ coefficient of }}{text{coefficient of }})
Sum of the products of two zeroes taken at a time
= (frac{ 1 }{ 2 } ) × 1 + (frac{ 1 }{ 2 } ) × (–2) + 1 × (–2)
= (frac{ 1 }{ 2 } ) – 1 – 2 = (frac{ -5 }{ 2 } ) = (frac{text{coefficient of }x}{text{coefficient of }})
Product of all the three zeroes
= (frac{ 1 }{ 2 } ) × (1) × (–2) = –1
= (frac{ -2 }{ 2 } ) = (frac{-text{ constant term }}{text{coefficient of }})
