Definitions [2]
The zeroes of a polynomial are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
Define a polynomial with real coefficients.
In the polynomial `f(x)=a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0`,
`a_nx^n,a_(n-1)x^(n-1),...,a_1x`, and `a_0` are known as the terms of the polynomial and `a_n,a_(n-1),...,a_1` and `a_0` are their real coefficients.
For example, `p(x)=3x-2` is a polynomial and 3 is a real coefficient.
Formulae [1]
| Polynomial | Roots | Sum of roots | Pairwise product | Product of roots |
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\[ax^2 + bx + c = 0\]
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\[\alpha, \beta\]
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\[-\frac{b}{a}\]
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Not applicable
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\[\frac{c}{a}\]
|
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\[ax^3 + bx^2 + cx + d = 0\]
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\[\alpha, \beta, \gamma\]
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\[-\frac{b}{a}\]
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\[\frac{c}{a}\]
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\[-\frac{d}{a}\]
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Theorems and Laws [2]
If α and β are the zeros of the quadratic polynomial f(x) = x2 – px + q, prove that `alpha^2/beta^2 + beta^2/alpha^2 = p^4/q^2 - (4p^2)/q + 2`.
Since α and β are the zeros of the quadratic polynomial f(x) = x2 – px + q
`alpha+beta=-"coefficient of x"/("coefficient of "x^2)`
`=(-(-p))/1`
= p
`alphabeta="constant term"/"coefficient of "x^2`
`=q/1`
= q
We have,
`alpha^2/beta^2+beta^2/alpha^2=(alpha^2xxalpha^2)/(beta^2xxalpha^2)+(beta^2xxbeta^2)/(alpha^2xxbeta^2)`
`alpha^2/beta^2+beta^2/alpha^2=alpha^4/(beta^2alpha^2)+beta^4/(alpha^2beta^2)`
`alpha^2/beta^2+beta^2/alpha^2=(alpha^4+beta^4)/(alpha^2beta^2)`
`alpha^2/beta^2+beta^2/alpha^2=((alpha^2+beta^2)-2alpha^2beta^2)/(alpha^2beta^2)`
`alpha^2/beta^2+beta^2/alpha^2=([(alpha+beta)^2-2alphabeta]^2-2(alphabeta)^2)/(alphabeta)^2`
`alpha^2/beta^2+beta^2/alpha^2=([(p)^2-2q]^2-2(q)^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=([p^2-2q]^2-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=([p^2xxp^2-2xxp^2xx2q+2qxx2q]-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=([p^4-4p^2q+4q^2]-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=(p^4-4p^2q+4q^2-2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=(p^4-4p^2q+2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2q)/q^2+(2q^2)/q^2`
`alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`
Hence, it is proved that `alpha^2/beta^2+beta^2/alpha^2" is equal to "p^4/q^2-(4p^2)/q+2`.
If the zeros of the polynomial f(x) = ax3 + 3bx2 + 3cx + d are in A.P., prove that 2b3 – 3abc + a2d = 0.
Let a - d, a and a + d be the zeros of the polynomial f(x). Then,
Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`
`a-d+a+a+d=(-3b)/a`
`3a=(-3b)/a`
`a=(-3b)/axx1/3`
`a=(-b)/a`
Since a is a zero of the polynomial f(x).
Therefore,
f(x) = ax3 + 3bx2 + 3cx + d
f(a) = 0
f(a) = aa3 + 3ba2 + 3ca + d
aa3 + 3ba2 + 3ca + d = 0
`a((-b)/a)^3+3bxx((-b)/a)^2+3cxx((-b)/a)+d=0`
`axx(-b)/axx(-b)/axx(-b)/a+3xxbxx(-b)/axx(-b)/a+3xxcxx(-b)/a+d=0`
`(-b^3)/a^2+(3b^3)/a^2-(3cb)/a+d=0`
`(-b^3+3b^3-3abc+a^2d)/a^2=0`
2b3 − 3abc + a2d = 0xa2
2b3 − 3abc + a2d = 0
Hence, it is proved that 2b3 − 3abc + a2d = 0.
Key Points
A quadratic polynomial can have only:
Case (i): Two zeroes
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Graph cuts the x-axis at two distinct points
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Two distinct real zeroes
Case (ii): One zero
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Graph touches the x-axis at exactly one point
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One real (repeated) zero
Case (iii): No zero
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The graph does not touch the x-axis at all
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No real zero
A polynomial of degree n can have at most n zeroes.
- For \[ ax^{2} + bx + c = 0: \]
\[ \alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a} \] - A quadratic with roots \[ \alpha, \beta \] is:
\[ x^{2} - (\alpha + \beta)x + \alpha\beta = 0 \] - For \[ ax^{3} + bx^{2} + cx + d = 0: \]
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]
\[ \alpha\beta\gamma = -\frac{d}{a} \] - Always identify coefficients only after writing the equation in standard form.
Important Questions [29]
- If α and β are the zeroes of the polynomial x2 – 1, then the value of (α + β) is ______.
- The zeroes of the polynomial 3x2 + 11x – 4 are ______.
- Find the Value of K Such that the Polynomial X2 − (K + 6)X + 2(2k −1) Has Sum of Its Zeros Equal to Half of Their Product.
- The number of polynomials having zeroes – 3 and 4 is ______.
- The graph of y = f(x) is shown in the figure for some polynomial f(x). The number of zeroes of f(x) is ______.
- The zeroes of the quadratic polynomial 16x2 – 9 are ______.
- Find the value of k such that the polynomial x2-(k +6)x+ 2(2k - 1) has some of its zeros equal to half of their product.
- If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is ______.
- If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and –3, then ______.
- The number of quadratic polynomials having zeroes –5 and –3 is ______.
- The graph of y = f(x) is shown in the figure for some polynomial f(x). The number of zeroes of f(x) are ______.
- If α and β are the zeroes of the polynomial x2 + x − 2, then find the value of αβ+βα
- If α, β are zeroes of the quadratic polynomial x2 – 5x + 6, form another quadratic polynomial whose zeroes are αβ1α,1β.
- Find a quadratic polynomial whose zeroes are 6 and – 3.
- Find the zeroes of the polynomial x2 + 4x – 12.
- The zeroes of the polynomial p(x) = 25x2 – 49 are ______.
- The zeroes of the polynomial p(x) = 2x2 – x – 3 are ______.
- If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then αβ(1α+1β) is equal to ______.
- Find All Zeroes of the Polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` If Two of Its Zeroes Are `(2 + Sqrt3)` and `(2 - Sqrt3)`
- Find the zeroes of the quadratic polynomial f(x) = 4x^2 – 4x – 3 and verify the relation between its zeroes and coefficients.
- Find the zeroes of the quadratic polynomial 4x^2 − 4x + 1 and verify the relation between the zeroes and the coefficients.
- Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x), where p(x) = x5 − 4x3 + x2 + 3x + 1, g(x) = x3 − 3x + 1
- If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
- Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
- Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients. 6x^2 – 3 – 7x
- A quadratic polynomial the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
- If p(x) = x2 + 5x + 6, then p(– 2) is ______.
- Find the zeroes of the quadratic polynomial x2 + 6x + 8 and verify the relationship between the zeroes and the coefficients.
- A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
