Polynomials Class 10 Notes for CBSE Board Exams

Key Points: Must Remember

• A polynomial $P(x)$ in one variable $x$ is an algebraic expression of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_0, a_1, ..., a_n$ are real numbers and $a_n \neq 0$, and $n$ is a non-negative integer (whole number).
• The degree of a polynomial is the highest power of the variable in the polynomial. E.g., Degree of $5x^3 - 2x + 7$ is 3.
• A zero of a polynomial $P(x)$ is a value 'k' such that $P(k) = 0$. Graphically, these are the x-intercepts.
• A linear polynomial ($ax+b$, degree 1) has exactly one zero.
• A quadratic polynomial ($ax^2+bx+c$, degree 2) has at most two zeros.
• A cubic polynomial ($ax^3+bx^2+cx+d$, degree 3) has at most three zeros.
• Relationship between Zeros and Coefficients for a quadratic polynomial $ax^2+bx+c$ (where $a \neq 0$): If $\alpha$ and $\beta$ are the zeros, then $\alpha + \beta = -b/a$ and $\alpha \beta = c/a$.
• Relationship between Zeros and Coefficients for a cubic polynomial $ax^3+bx^2+cx+d$ (where $a \neq 0$): If $\alpha, \beta, \gamma$ are the zeros, then $\alpha + \beta + \gamma = -b/a$, $\alpha\beta + \beta\gamma + \gamma\alpha = c/a$, and $\alpha\beta\gamma = -d/a$.
• Division Algorithm for Polynomials: If $P(x)$ and $G(x)$ are any two polynomials with $G(x) \neq 0$, then we can find polynomials $Q(x)$ and $R(x)$ such that $P(x) = G(x) \cdot Q(x) + R(x)$, where $R(x) = 0$ or degree of $R(x) <$ degree of $G(x)$. (Dividend = Divisor \times Quotient + Remainder).

Key Definitions

Polynomial

An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Degree of a Polynomial

The highest power of the variable in a polynomial expression.

Zero of a Polynomial

A value of the variable for which the polynomial evaluates to zero. Graphically, it's the x-coordinate where the graph intersects the x-axis.

Linear Polynomial

A polynomial of degree 1, generally of the form $ax+b$, where $a \neq 0$.

Quadratic Polynomial

A polynomial of degree 2, generally of the form $ax^2+bx+c$, where $a \neq 0$.

Cubic Polynomial

A polynomial of degree 3, generally of the form $ax^3+bx^2+cx+d$, where $a \neq 0$.

Coefficient

A numerical or constant quantity placed before and multiplying the variable in an algebraic term (e.g., in $5x^2$, 5 is the coefficient of $x^2$).

Monomial

A polynomial with only one term (e.g., $5x^3$).

Binomial

A polynomial with two terms (e.g., $2x+7$).

Trinomial

A polynomial with three terms (e.g., $x^2-3x+5$).

Understanding Zeros and Their Relationship with Coefficients

The zeros of a polynomial are the fundamental values for which the polynomial's expression evaluates to zero. Geometrically, if you plot the graph of a polynomial, its zeros correspond to the points where the graph intersects the x-axis. For a linear polynomial, say $ax+b$, there's always one unique zero, which is $x = -b/a$. For example, for $2x-4$, the zero is $x=2$.

Things get more interesting with quadratic polynomials of the form $ax^2+bx+c$, where $a \neq 0$. A quadratic polynomial can have at most two zeros. These zeros can be found by setting $ax^2+bx+c = 0$ and solving for $x$, typically using factorization or the quadratic formula. If $\alpha$ and $\beta$ are the two zeros, there's a powerful relationship between these zeros and the coefficients of the polynomial:

• Sum of Zeros ($\alpha + \beta$): This is always equal to $-b/a$.
• Product of Zeros ($\alpha \beta$): This is always equal to $c/a$.

These relationships are incredibly useful for forming a quadratic polynomial when its zeros (or sum/product of zeros) are given, or for verifying the zeros. For instance, if you are given zeros 2 and 3, the sum is 5 and product is 6. A quadratic polynomial could be $x^2 - (sum)x + (product) = x^2 - 5x + 6$.

Similarly, for cubic polynomials of the form $ax^3+bx^2+cx+d$, where $a \neq 0$, there can be at most three zeros. Let these zeros be $\alpha, \beta$, and $\gamma$. The relationships are extended:

• Sum of Zeros ($\alpha + \beta + \gamma$): This equals $-b/a$.
• Sum of the products of the zeros taken two at a time ($\alpha\beta + \beta\gamma + \gamma\alpha$): This equals $c/a$.
• Product of Zeros ($\alpha\beta\gamma$): This equals $-d/a$.

Understanding these relationships is crucial for solving problems where zeros are related to coefficients or for constructing polynomials given their zeros. Remember, the coefficients $a, b, c, d$ are critical in these formulas, especially the leading coefficient $a$ which appears in the denominator.

Polynomial Division Algorithm (Long Division)

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Worked Examples

• {"heading":"Example 1: Zeros & Coefficients (Quadratic)","bodyMarkdown":"Q: Find the zeros of the quadratic polynomial $x^2 - 2x - 8$ and verify the relationship between the zeros and coefficients.\n\nA:\n1. Find zeros: Factorize $x^2 - 2x - 8 = (x-4)(x+2)$. Setting each factor to zero gives $x=4$ and $x=-2$. So, $\\alpha = 4, \\beta = -2$.\n2. Identify coefficients: For $x^2 - 2x - 8$, $a=1, b=-2, c=-8$.\n3. Verify relationships:\n Sum of zeros: $\\alpha + \\beta = 4 + (-2) = 2$. From formula: $-b/a = -(-2)/1 = 2$. (Verified)\n Product of zeros: $\\alpha \\beta = 4 \\times (-2) = -8$. From formula: $c/a = -8/1 = -8$. (Verified)"}
• {"heading":"Example 2: Forming Quadratic Polynomial","bodyMarkdown":"Q: Find a quadratic polynomial whose sum and product of zeros are $-3$ and $2$, respectively.\n\nA:\nGiven $\\alpha + \\beta = -3$ and $\\alpha \\beta = 2$.\nA quadratic polynomial can be written as $k[x^2 - (\\alpha + \\beta)x + \\alpha \\beta]$.\nSubstitute the given values: $k[x^2 - (-3)x + 2] = k[x^2 + 3x + 2]$.\nFor $k=1$, the polynomial is $x^2 + 3x + 2$. (Any non-zero real value of $k$ gives a valid polynomial)."}

Exam Tips & Common Mistakes

1. Sign Errors: Be extremely careful with negative signs, especially when calculating $-b/a$ or $-d/a$. A common mistake is to forget the negative sign or misapply it.
2. Missing Terms in Division: When performing polynomial long division, if a power of $x$ is missing in the dividend, represent it with a zero coefficient (e.g., $x^3+5$ becomes $x^3+0x^2+0x+5$). This helps maintain proper alignment during subtraction and prevents errors.
3. Verification is Key: After finding the zeros of a quadratic or cubic polynomial, always take a moment to verify the relationships with coefficients. This simple step can catch calculation errors and ensure full marks.
4. Graphical Interpretation: Remember that the number of zeros a polynomial has is at most equal to its degree. For linear and quadratic polynomials, the number of zeros corresponds to the number of times the graph intersects the x-axis.

Quick Revision Check

• Q: What is the degree of the polynomial $P(x) = 7x^5 - 3x^2 + \sqrt{2}x - 1/2$? A: The highest power of $x$ is 5, so the degree is 5.
• Q: If $\alpha$ and $\beta$ are the zeros of the polynomial $x^2 + 5x + 6$, find the value of $\alpha + \beta$. A: For $ax^2+bx+c$, sum of zeros $= -b/a$. Here, $a=1, b=5, c=6$. So, $\alpha + \beta = -5/1 = -5$.
• Q: Can a polynomial have negative exponents for its variable? A: No, by definition, the exponents of the variable in a polynomial must be non-negative integers (whole numbers).
• Q: What is the significance of the remainder being zero after polynomial division? A: If the remainder is zero, it means the divisor is a factor of the dividend, and the quotient is also a factor.

Frequently Asked Questions