Tenth Grade Mathematics: Relationship Between Zeros and Coefficients

Algebraic equations possess a remarkable symmetry: the roots of a polynomial are completely determined by its coefficients, and conversely, the coefficients can be expressed as symmetric polynomials of its roots. Discovered systematically by the French mathematician François Viète, these relationships—known collectively as Vieta's formulas—allow us to verify calculated zeros, determine unknown coefficients, and construct new polynomials without performing extensive expansions.

The Quadratic Relationship

Let P(x) = ax^2 + bx + c be a quadratic polynomial where a != 0. Let alpha and beta denote the two zeros of P(x).

Factor Theorem Correspondence:
If alpha and beta are zeros of P(x), then (x - alpha) and (x - beta) are linear factors:
P(x) = k * (x - alpha) * (x - beta)

where k is a non-zero real constant. Expanding this product:

P(x) = k * [x^2 - (alpha + beta)*x + alpha * beta]
P(x) = k*x^2 - k*(alpha + beta)*x + k*(alpha * beta)

Now equate this with the standard quadratic polynomial ax^2 + bx + c by comparing coefficients of identical powers of x:

Coefficient of x^2:  a = k
Coefficient of x^1:  b = -k * (alpha + beta)  ===>  b = -a * (alpha + beta)
Constant term:       c = k * (alpha * beta)   ===>  c = a * (alpha * beta)

Dividing both equations by a yields the celebrated fundamental relationships for quadratic polynomials:

1. Sum of Zeros:
   alpha + beta = -b / a = - (Coefficient of x) / (Coefficient of x^2)

2. Product of Zeros:
   alpha * beta = c / a  = (Constant term) / (Coefficient of x^2)

Verification Example: P(x) = x^2 - 2*x - 8

To demonstrate the application of these formulas, consider finding the zeros and verifying the coefficient relationships for P(x) = x^2 - 2*x - 8.

Step 1: Find the zeros by factoring:
        x^2 - 2*x - 8 = x^2 - 4*x + 2*x - 8
                      = x*(x - 4) + 2*(x - 4)
                      = (x - 4)*(x + 2)
Setting P(x) = 0 gives zeros: alpha = 4, beta = -2.

Step 2: Identify coefficients:
        a = 1, b = -2, c = -8.

Step 3: Verify Sum of Zeros:
        alpha + beta = 4 + (-2) = 2
        Formula: -b / a = -(-2) / 1 = 2
        Verified: alpha + beta = -b / a.

Step 4: Verify Product of Zeros:
        alpha * beta = 4 * (-2) = -8
        Formula: c / a = -8 / 1 = -8
        Verified: alpha * beta = c / a.

The Cubic Relationship

Vieta's formulas extend seamlessly to degree-three polynomials. Let P(x) = ax^3 + bx^2 + c*x + d be a cubic polynomial (a != 0) with zeros alpha, beta, and gamma.

Factored Form:
P(x) = a * (x - alpha) * (x - beta) * (x - gamma)
     = a * [x^3 - (alpha + beta + gamma)*x^2 + (alpha*beta + beta*gamma + gamma*alpha)*x - alpha*beta*gamma]

Comparing coefficients with ax^3 + bx^2 + c*x + d gives the three fundamental cubic relations:

1. Sum of Zeros:
   alpha + beta + gamma = -b / a

2. Sum of Products of Zeros taken two at a time:
   alpha*beta + beta*gamma + gamma*alpha = c / a

3. Product of all Zeros:
   alpha * beta * gamma = -d / a

Notice the alternating sign pattern: the first sum carries a negative sign (-b/a), the second carries a positive sign (+c/a), and the third carries a negative sign (-d/a). This alternating rule holds for polynomials of any degree n.

Evaluating Symmetric Functions of Zeros

Often, examination problems require calculating expressions involving zeros—such as alpha^2 + beta^2 or 1/alpha + 1/beta—without calculating the numerical values of alpha and beta individually. We achieve this by rewriting the expressions in terms of the elementary symmetric sums (alpha + beta) and (alpha * beta).

Common Symmetric Transformations:
1. alpha^2 + beta^2 = (alpha + beta)^2 - 2*alpha*beta
2. (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta
3. 1/alpha + 1/beta = (alpha + beta) / (alpha * beta)
4. alpha^3 + beta^3 = (alpha + beta)^3 - 3*alpha*beta*(alpha + beta)

Worked Example:
If alpha and beta are zeros of 2*x^2 - 5*x + 7, find the value of (1/alpha + 1/beta).
Coefficients: a = 2, b = -5, c = 7.
Sum of zeros:     alpha + beta = -(-5)/2 = 5/2.
Product of zeros: alpha * beta = 7/2.
Value of 1/alpha + 1/beta = (alpha + beta) / (alpha * beta)
                          = (5/2) / (7/2) = 5/7.

Chapter Exercises

Exercise 1: Find the zeros of the quadratic polynomial 6x^2 - 3 - 7x, and verify the relationship between the zeros and the coefficients.

Exercise 2: If alpha and beta are the zeros of the quadratic polynomial P(x) = x^2 - 4*x + 3, calculate the exact values of: (a) alpha^2 + beta^2, (b) alpha/beta + beta/alpha, and (c) alpha^3 + beta^3.

Exercise 3: Verify that 3, -1, and -1/3 are the zeros of the cubic polynomial P(x) = 3x^3 - 5x^2 - 11*x - 3, and then verify the relationships between its zeros and coefficients.

Solutions and Explanations

Solution 1: First, write the polynomial in standard form: P(x) = 6x^2 - 7x - 3. To find the zeros, factor by splitting the middle term: we need two numbers whose product is 6 * (-3) = -18 and whose sum is -7; these are -9 and 2. 6x^2 - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (2x - 3)(3*x + 1). Setting each factor to zero yields alpha = 3/2 and beta = -1/3. Coefficients are a = 6, b = -7, c = -3. Sum of zeros: alpha + beta = 3/2 + (-1/3) = 9/6 - 2/6 = 7/6. Formula: -b/a = -(-7)/6 = 7/6 (Verified). Product of zeros: alpha * beta = (3/2) * (-1/3) = -3/6 = -1/2. Formula: c/a = -3/6 = -1/2 (Verified).

Solution 2: For P(x) = x^2 - 4x + 3, the coefficients are a = 1, b = -4, c = 3. The sum of zeros is alpha + beta = -(-4)/1 = 4, and the product of zeros is alpha * beta = 3/1 = 3. For part (a): alpha^2 + beta^2 = (alpha + beta)^2 - 2alphabeta = 4^2 - 2(3) = 16 - 6 = 10. For part (b): alpha/beta + beta/alpha = (alpha^2 + beta^2) / (alpha * beta) = 10 / 3. For part (c): alpha^3 + beta^3 = (alpha + beta)^3 - 3alphabeta(alpha + beta) = 4^3 - 3(3)*(4) = 64 - 36 = 28.

Solution 3: We test each candidate zero in P(x) = 3x^3 - 5x^2 - 11x - 3. P(3) = 3(27) - 5(9) - 11(3) - 3 = 81 - 45 - 33 - 3 = 0. P(-1) = 3(-1) - 5(1) - 11(-1) - 3 = -3 - 5 + 11 - 3 = 0. P(-1/3) = 3(-1/27) - 5(1/9) - 11(-1/3) - 3 = -1/9 - 5/9 + 33/9 - 27/9 = 0. Hence, alpha = 3, beta = -1, gamma = -1/3 are confirmed zeros. Coefficients: a = 3, b = -5, c = -11, d = -3. Sum: alpha + beta + gamma = 3 - 1 - 1/3 = 5/3; Formula: -b/a = -(-5)/3 = 5/3 (Verified). Sum of pairwise products: alphabeta + betagamma + gammaalpha = 3(-1) + (-1)(-1/3) + (-1/3)(3) = -3 + 1/3 - 1 = -11/3; Formula: c/a = -11/3 (Verified). Product: alphabetagamma = 3 * (-1) * (-1/3) = 1; Formula: -d/a = -(-3)/3 = 1 (Verified).