CBSE Class 10 Maths: Polynomials Exercise 2.2 – Zeros and Coefficients

The Fundamental Relationship: Zeros and Coefficients

A quadratic polynomial is generally expressed in the form ax² + bx + c, where a, b, and c are real numbers and a ≠ 0. The 'zeros' of a polynomial are the values of x for which the polynomial equals zero. For a quadratic polynomial, there are at most two zeros. Let's denote these zeros as α (alpha) and β (beta).

There's a beautiful and consistent relationship between these zeros and the coefficients a, b, and c:

1. Sum of Zeros (α + β): The sum of the zeros of a quadratic polynomial ax² + bx + c is always equal to the negative of the coefficient of x divided by the coefficient of x².

Mathematically, α + β = -b/a.

1. Product of Zeros (αβ): The product of the zeros of a quadratic polynomial ax² + bx + c is always equal to the constant term divided by the coefficient of x².

Mathematically, αβ = c/a.

These relationships are incredibly useful. They allow us to quickly check our answers when finding zeros or even to construct a quadratic polynomial if we know its zeros or their sum and product. This concept is vital for understanding polynomial behavior and solving more complex algebraic problems. Remember, if a quadratic polynomial has zeros α and β, it can also be written in the form k(x - α)(x - β), where k is any non-zero real number. Expanding this gives k[x² - (α + β)x + αβ], which when compared to ax² + bx + c, helps us understand why these relationships hold true.

Key Definitions for Polynomials Ex 2.2

Polynomial

An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, 3x² + 2x - 5.

Quadratic Polynomial

A polynomial of degree 2, meaning the highest power of the variable is 2. Its general form is ax² + bx + c, where a ≠ 0.

Zeros of a Polynomial

The values of the variable (e.g., x) for which the polynomial evaluates to zero. These are also known as roots of the polynomial.

Coefficients

The numerical or constant part of a term in an algebraic expression. In ax² + bx + c, a, b, and c are the coefficients (where c is the constant term).

Worked Examples: Finding Zeros and Verifying Relationships

• Example 1: Find the zeros of the quadratic polynomial x² + 7x + 10 and verify the relationship between the zeros and the coefficients. Step 1: Set the polynomial to zero to find the zeros. x² + 7x + 10 = 0 Step 2: Factorize the quadratic polynomial. We need two numbers that multiply to 10 and add to 7. These are 5 and 2. x² + 5x + 2x + 10 = 0 x(x + 5) + 2(x + 5) = 0 (x + 5)(x + 2) = 0 Step 3: Solve for x to find the zeros. x + 5 = 0 => x = -5 x + 2 = 0 => x = -2 So, the zeros are α = -5 and β = -2. Step 4: Identify the coefficients a, b, c from x² + 7x + 10. a = 1, b = 7, c = 10. Step 5: Verify the sum of zeros. α + β = -5 + (-2) = -7 -b/a = -(7)/1 = -7 Since -7 = -7, the relationship is verified. Step 6: Verify the product of zeros. αβ = (-5) * (-2) = 10 c/a = 10/1 = 10 Since 10 = 10, the relationship is verified.
• Example 2: Find the zeros of the polynomial 4u² + 8u and verify the relationship between the zeros and the coefficients. Step 1: Set the polynomial to zero. 4u² + 8u = 0 Step 2: Factorize the polynomial (common factor). 4u(u + 2) = 0 Step 3: Solve for u to find the zeros. 4u = 0 => u = 0 u + 2 = 0 => u = -2 So, the zeros are α = 0 and β = -2. Step 4: Identify the coefficients a, b, c from 4u² + 8u. (Remember c=0 for missing constant term). a = 4, b = 8, c = 0. Step 5: Verify the sum of zeros. α + β = 0 + (-2) = -2 -b/a = -(8)/4 = -2 Since -2 = -2, the relationship is verified. Step 6: Verify the product of zeros. αβ = (0) * (-2) = 0 c/a = 0/4 = 0 Since 0 = 0, the relationship is verified.
• Example 3: Find a quadratic polynomial whose sum and product of zeros are -3 and 2, respectively. Step 1: Recall the general form of a quadratic polynomial given its sum and product of zeros: k(x² - (sum of zeros)x + (product of zeros)) Step 2: Substitute the given values. Sum of zeros (α + β) = -3 Product of zeros (αβ) = 2 Polynomial = k(x² - (-3)x + 2) Polynomial = k(x² + 3x + 2) Step 3: Choose a suitable value for k. For simplicity, we usually take k=1 unless specified otherwise. If k=1, the polynomial is x² + 3x + 2. Final Answer: A quadratic polynomial is x² + 3x + 2.

Exam Tip: Avoiding Common Mistakes in Ex 2.2

Many students make small errors that cost valuable marks in exams. Here are a few to watch out for:

1. Sign Errors in -b/a: Always be careful with the negative sign in the sum of zeros formula. If b is already negative, then -b will become positive. For example, if the polynomial is x² - 5x + 6, then b = -5, so -b/a = -(-5)/1 = 5.
2. Incorrectly Identifying Coefficients: Ensure you correctly identify a, b, and c from the polynomial. Sometimes the terms might not be in standard order (e.g., 6x² - 7x - 3 instead of 6x² - 3 - 7x). Rearrange it to ax² + bx + c before picking coefficients.
3. Forgetting a in the Denominator: Remember that a is in the denominator for both formulas (-b/a and c/a). If a is not 1, this factor is crucial. For example, for 2x² + 3x + 1, a=2, b=3, c=1. Sum is -3/2, product is 1/2.
4. Leaving k out when forming polynomials: When asked to form a polynomial given the sum and product of zeros, the general form is k(x² - (sum)x + product). While k=1 is often used for simplicity, it's good practice to acknowledge k or state that you're choosing the simplest polynomial where k=1.

Practice Questions with Solutions

• Q: Find the zeros of the quadratic polynomial x² - 2x - 8 and verify the relationship between the zeros and the coefficients. A: Step 1: Set the polynomial to zero: x² - 2x - 8 = 0. Step 2: Factorize: x² - 4x + 2x - 8 = 0 => x(x - 4) + 2(x - 4) = 0 => (x - 4)(x + 2) = 0. Step 3: Zeros are x = 4 and x = -2 (so α = 4, β = -2). Step 4: Identify coefficients: a = 1, b = -2, c = -8. Step 5: Verify sum: α + β = 4 + (-2) = 2. Also, -b/a = -(-2)/1 = 2. Verified. Step 6: Verify product: αβ = 4 * (-2) = -8. Also, c/a = -8/1 = -8. Verified. Final answer: Zeros are 4 and -2. Relationships are verified.
• Q: Find the zeros of 6x² - 7x - 3 and verify the relationship between the zeros and the coefficients. A: Step 1: Set 6x² - 7x - 3 = 0. Step 2: Factorize (by splitting the middle term: product = 6(-3)=-18, sum = -7; numbers are -9, 2): 6x² - 9x + 2x - 3 = 0 => 3x(2x - 3) + 1(2x - 3) = 0 => (2x - 3)(3x + 1) = 0. Step 3: Zeros are x = 3/2 and x = -1/3 (so α = 3/2, β = -1/3). Step 4: Coefficients: a = 6, b = -7, c = -3. Step 5: Verify sum: α + β = 3/2 + (-1/3) = (9 - 2)/6 = 7/6. Also, -b/a = -(-7)/6 = 7/6. Verified. Step 6: Verify product: αβ = (3/2) (-1/3) = -3/6 = -1/2. Also, c/a = -3/6 = -1/2. Verified. Final answer: Zeros are 3/2 and -1/3. Relationships are verified.
• Q: Find a quadratic polynomial, the sum and product of whose zeros are 1/4 and -1 respectively. A: Step 1: Recall the general form: k(x² - (sum of zeros)x + (product of zeros)). Step 2: Substitute the given sum (1/4) and product (-1). k(x² - (1/4)x + (-1)) k(x² - x/4 - 1) Step 3: To eliminate the fraction, let k=4 (a common denominator). 4(x² - x/4 - 1) 4x² - x - 4 Final answer: A quadratic polynomial is 4x² - x - 4.
• Q: Find the zeros of t² - 15 and verify the relationship between the zeros and coefficients. A: Step 1: Set t² - 15 = 0. Step 2: Factorize using a² - b² = (a - b)(a + b): t² - (√15)² = 0 => (t - √15)(t + √15) = 0. Step 3: Zeros are t = √15 and t = -√15 (so α = √15, β = -√15). Step 4: Coefficients: a = 1, b = 0, c = -15 (since there's no t term). Step 5: Verify sum: α + β = √15 + (-√15) = 0. Also, -b/a = -0/1 = 0. Verified. Step 6: Verify product: αβ = (√15) * (-√15) = -15. Also, c/a = -15/1 = -15. Verified. Final answer: Zeros are √15 and -√15. Relationships are verified.

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