Polynomials:
The chapter 'Polynomial' carrying the weightage of \(4\) marks in the board examination. It covers the concepts like zeroes of a polynomial, relationship between zeroes and coefficients and the geometrical meaning of zeroes.
Most possible variation of question types that we can expect in board exam as per previous year question paper are discussed below.
| Total Marks \(4\) | |
| Variation \(1\) | Variation \(2\) |
Total Mark \(= 4\)
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Total Mark \(= 4\)
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To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Zeroes of a polynomial - Finding zeroes, Finding quadratic polynomial using zeroes
- Relationship between zeroes and coefficients - Sum and product of zeroes, Verifying the relation
- Geometrical meaning of zeroes - Finding number of zeros using graphs.
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
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Zeroes of a polynomial
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Sec A - Assertion Reasoning,
Sec B
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Relationship between zeroes and coefficient
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Sec A, Sec C | |
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Geometrical Meaning of
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Sec A |
Let us recall the concepts in Polynomial:
1. Polynomials of degrees \(1\), \(2\) and \(3\) are called linear, quadratic and cubic polynomials respectively. A quadratic polynomial in \(x\) with real coefficients is of the form \(ax^2 + bx + c\), where \(a\), \(b\), \(c\) are real numbers with a \(0\). A quadratic polynomial can have at most \(2\) zeroes and a cubic polynomial can have at most \(3\) zeroes.
2. The zeroes of a polynomial \(p(x)\) are precisely the \(x\)-coordinates of the points, where the graph of \(y = p(x)\) intersects the \(x\) -axis.
3. If \(\alpha \) is the zero of the linear polynomial \(p(x)=ax+b\) then
\(\alpha = \frac{-b}{a}\) \(= \frac{-\text{constant term}}{\text{coefficient of }x} \)
4. If \(\alpha, \beta\) is the zero of the linear polynomial \(p(x)=ax^2+bx+c\) then
\(\alpha+\beta = \frac{-b}{a} = \frac{-\text{coefficient of }x}{\text{coefficient of }x^2}\)
\(\alpha\beta = \frac{c}{a} = \frac{\text{constant term}}{\text{coefficient of }x^2}\)
The general form of the quadratic polynomial based on its zeros is given by
\(p(x)=x^2-(\text{sum of zeroes})x+\text{product of zeroes}\)
\(p(x)=x^2-(\alpha+\beta)x+\alpha\beta\).
5. If \(\alpha,\beta,\gamma\) is the zero of the linear polynomial \(p(x)=ax^3+bx^2+cx+d\) then
\(\alpha+\beta+\gamma = \frac{-b}{a} = \frac{-\text{coefficient of }x^2}{\text{coefficient of }x^3}\)
\(\alpha\beta+\beta\gamma+\gamma\alpha = \frac{c}{a} = \frac{\text{coefficient of }x}{\text{coefficient of }x^2}\)
\(\alpha\beta\gamma = \frac{-d}{a} = \frac{-\text{constant term}}{\text{coefficent of }x^3}\)
The general form of the cubic polynomial based on its zeros is given by
\(p(x)=x^3-(\text{sum of zeroes})x^2+(\text{taking two roots at a time})x-\text{product of zeroes}\)
\(p(x)=x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x- \alpha\beta\gamma\).
Commom mistakes to avoid:
1. Counting the intersection with both axes as zeros, for the polynomial \(p(x)\), the zeros are only the points where its graph intersects or touches the \(x\)-axis.
2. Confusing between the factor and zero: if \(p(a) = 0\), then \(a\) is a zero and \((x-a)\) is a factor.
3. Making sign erros while solving a quadratic polynomial. While writing the sum of zeros the formula is \(\alpha + \beta = \frac{-b}{a}\) not \(\frac{b}{a}\).
4. Forgetting the negative sign in quadratic polynomial formation, the general form is \(x^2 - (\alpha + \beta )x + \alpha\beta\).
5. Calculation errors while checking relation between the coefficients and zeros of the polynomial. Need to verify the obtained zeros satisfy the required sum/product relationships.
Exam tips & tricks:
1. Number of zeros of the polynomial is equal to the maximum degree of the polynomial.
For example: A quadratic polynomial can have at most \(2\) zeros, it can have \(2\), \(1\) or \(0\) as real zeros.
2. Coefficient extraction: Write \(a\), \(b\) and \(c\) explicitly with their signs before applying formulas.
3. Cross-check your zeros by factoring and back substitution into the relation.
4. For values like \(\frac{1}{{\alpha}^2} + \frac{1}{{\beta}^2}\), simplify the algebraic fractions first.