Polynomials (Class 10): Notes और MCQ Practice

आसान भाषा में:

Polynomial एक ऐसा algebraic expression है जिसमें variable की power हमेशा एक whole number होती है। Class 10 में हमारा मुख्य focus Quadratic Polynomial (ax² + bx + c) पर है। Zeroes variable (x) की वो values होती हैं जिन्हें polynomial में रखने पर पूरा answer 0 बन जाए। Geometrically, polynomial का graph x-axis को जितनी बार काटता है, उतने ही उसके zeroes होते हैं।

Key Formulas & Concepts:

• Linear Polynomial: ax + b (Degree = 1, Max zeroes = 1, Zero = -b/a)
• Quadratic Polynomial: ax² + bx + c (Degree = 2, Shape = Parabola, Max zeroes = 2)
• Sum of zeroes (α + β): -b ÷ a = -(Coefficient of x) ÷ (Coefficient of x²)
• Product of zeroes (α × β): c ÷ a = Constant term ÷ (Coefficient of x²)
• Forming Quadratic Polynomial: जब zeroes दिए हों:

k[x² − (α + β)x + αβ] या k[x² − (Sum)x + Product]

• Graph Rule: y = p(x) के लिए graph x-axis को जितने points पर intersect करता है, उतने real zeroes होते हैं।

2 Super Memory Tricks:

1. Sign Confusion खत्म: Sum of zeroes में हमेशा Subtraction sign आता है (-b/a), जबकि Product में Positive sign रहता है (+c/a)।
2. Equation बनाने की Trick: x² − Sx + P याद रखें (पहले Subtract करो Sum को, फिर Plus करो Product को)।

2 Common Board Exam Mistakes:

1. Double Negative की गलती: Sum of zeroes निकालते समय formula का minus भूल जाना; अगर b = -5 है, तो Sum = -(-5) ÷ a = 5/a होगा।
2. y-axis का intersection गिनना: Graph वाले question में केवल x-axis के cut points गिनने होते हैं, गलती से y-axis के cut point को मत जोड़ लेना!

Class 10 Mathematics: Polynomials

1. Chapter in Short

This chapter deals with polynomials, focusing primarily on quadratic polynomials (polynomials of degree 2). It teaches students:

• How to find the "zeroes" (roots) of a polynomial.
• What zeroes look like on a graph (the points where the curve crosses the x-axis).
• The direct relationship between the zeroes of a quadratic polynomial and its coefficients.
• How to reconstruct a quadratic polynomial when the sum and product of its zeroes are known.

2. Key Concepts Explained Simply

What is a Polynomial and its Degree?

A polynomial is an algebraic expression made of variables, numbers, and exponents (which must be non-negative whole numbers).

• The degree is the highest power of the variable in the polynomial.
• Linear Polynomial: Degree 1 (e.g., 2x + 3)
• Quadratic Polynomial: Degree 2 (e.g., x² − 4x + 4)
• Cubic Polynomial: Degree 3 (e.g., x³ − 2x² + x − 1)

What is a "Zero" of a Polynomial?

A zero of a polynomial is simply a value of the variable that makes the whole expression equal to 0.

• Everyday Example: Think of a bank account balance represented by the formula Balance = 500 − 50d, where d is the number of days. The day your balance hits 0 (here, d = 10) is the "zero" of that formula.

Geometrical Meaning of Zeroes

When you plot a polynomial y = p(x) on a graph:

• The zeroes are the x-coordinates of the points where the graph intersects or touches the x-axis.
• If a graph cuts the x-axis at 2 points, it has 2 zeroes.
• If it cuts the x-axis at 1 point (or just touches it), it has 1 zero.
• If it does not touch the x-axis at all, it has 0 real zeroes.
• A quadratic polynomial graph forms a U-shaped curve called a parabola. It opens upwards if the coefficient of x² is positive (a > 0), and downwards if it is negative (a < 0).

Relationship Between Zeroes and Coefficients

For any quadratic polynomial ax² + bx + c (where a ≠ 0), let the zeroes be called α (Alpha) and β (Beta).

• Adding the zeroes together gives: α + β = −b ÷ a
• Multiplying the zeroes together gives: α × β = c ÷ a

3. Important Definitions, Formulas, and Facts

• General form of a quadratic polynomial:

p(x) = ax² + bx + c, where a, b, c are real numbers and a ≠ 0.

• Maximum zeroes rule: A polynomial of degree n can have at most n real zeroes.
• Linear: at most 1 zero.
• Quadratic: at most 2 zeroes.
• Cubic: at most 3 zeroes.
• Sum of zeroes (Quadratic):

α + β = −(coefficient of x) ÷ (coefficient of x²) = −b ÷ a

• Product of zeroes (Quadratic):

α × β = (constant term) ÷ (coefficient of x²) = c ÷ a

• Forming a Quadratic Polynomial:

If α and β are the zeroes, the polynomial is:

k [x² − (α + β)x + (α × β)]

or in words:

k [x² − (Sum of zeroes)x + (Product of zeroes)], where k is any non-zero real constant.

4. Solved Examples (Board-Exam Style)

Example 1: Finding Zeroes from a Graph

Question: The graph of y = p(x) intersects the x-axis at 3 distinct points and intersects the y-axis at 1 point. How many zeroes does p(x) have?

• Answer: The polynomial has 3 zeroes.
• Reason: The number of zeroes of p(x) depends only on the number of points where the graph intersects the x-axis, regardless of where it cuts the y-axis.

Example 2: Find Zeroes and Verify the Relationship

Question: Find the zeroes of the quadratic polynomial x² − 2x − 8 and verify the relationship between the zeroes and the coefficients.

Solution:

1. Factorize using splitting the middle term:

x² − 2x − 8 = x² − 4x + 2x − 8

= x(x − 4) + 2(x − 4)

= (x − 4)(x + 2)

1. Set p(x) = 0:

x − 4 = 0 or x + 2 = 0

x = 4 or x = −2

So, the zeroes are α = 4 and β = −2.

1. Compare x² − 2x − 8 with ax² + bx + c:

a = 1, b = −2, c = −8

1. Verification:

• Sum of zeroes: α + β = 4 + (−2) = 2

Formula: −b ÷ a = −(−2) ÷ 1 = 2

(Verified: α + β = −b ÷ a)

• Product of zeroes: α × β = 4 × (−2) = −8

Formula: c ÷ a = −8 ÷ 1 = −8

(Verified: α × β = c ÷ a)

Example 3: Rearranging Before Factorization

Question: Find the zeroes of the polynomial 6x² − 3 − 7x.

Solution:

1. Write in standard order ax² + bx + c:

p(x) = 6x² − 7x − 3

1. Factorize by splitting the middle term (product = 6 × (−3) = −18, sum = −7):

6x² − 9x + 2x − 3

= 3x(2x − 3) + 1(2x − 3)

= (2x − 3)(3x + 1)

1. Set each factor to 0:

2x − 3 = 0 => x = 3/2

3x + 1 = 0 => x = −1/3

Zeroes: 3/2 and −1/3.

Example 4: Creating a Quadratic Polynomial

Question: Find a quadratic polynomial whose sum and product of zeroes are 1/4 and −1 respectively.

Solution:

1. Let the zeroes be α and β.

• Sum: α + β = 1/4
• Product: α × β = −1

1. Use the standard formula:

p(x) = k [x² − (Sum)x + (Product)]

p(x) = k [x² − (1/4)x + (−1)]

p(x) = k [x² − (1/4)x − 1]

1. Choose k = 4 to clear the fraction:

p(x) = 4(x² − (1/4)x − 1) = 4x² − x − 4

Answer: A quadratic polynomial is 4x² − x − 4.

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid:

• Y-axis Trap: In graphical questions, students often count the points where the curve crosses the y-axis. Remember: Only count intersections with the x-axis for zeroes of y = p(x).
• Sign Error in Sum of Zeroes: Forgetting the negative sign in the sum formula. Remember, Sum = −b ÷ a, not b ÷ a.
• Unordered Terms: Identifying coefficients without arranging terms in decreasing power. For example, in 7x − 3 + 2x², a is 2, not 7!
• Sign Flip in Polynomial Formation: The formula is x² − (Sum)x + (Product). Notice the minus sign before the sum and the plus sign before the product.

Quick Revision Points:

• Degree 1 = Linear (1 zero max), Degree 2 = Quadratic (2 zeroes max), Degree 3 = Cubic (3 zeroes max).
• For ax² + bx + c:
• Sum = −b ÷ a
• Product = c ÷ a
• Given sum S and product P, the polynomial is x² − Sx + P.