Definitions [3]
The highest power of the variable in a polynomial is called its degree.
A real number k is a zero of p(x) if p(k) = 0.
The zeroes of a polynomial are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
Formulae [3]
For
p(x) = ax + b
Zero:
For a quadratic polynomial
ax2 + bx + c (a≠0)
If its zeroes are α and β, then:
\[\alpha+\beta=-\frac{b}{a}\]
\[\alpha\beta=\frac{c}{a}\]
For a cubic polynomial
ax3 + bx2 + cx + d,
\[\alpha+\beta+\gamma=-\frac{b}{a}\]
\[\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}\]
\[\alpha\beta\gamma=-\frac{d}{a}\]
Key Points
Quadratic polynomial
ax2 + bx + c
Cubic polynomial
ax3 + bx2 + cx + d
A quadratic polynomial can have only:
Case (i): Two zeroes
-
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.
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) = 4x2 - 4x - 3 and verify the relation between its zeroes and coefficients.
- Find the zeroes of the quadratic polynomial 4x2-4x+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 ______.
