Comparing x² - 2x - 8 with the standard quadratic polynomial form ax² + bx + c, we get the coefficients a = 1, b = -2, and c = -8.

The sum of the zeroes is 4 + (-2) = 2. Calculating -(b)/(a) gives -(-2)/(1) = 2. Since both expressions yield 2, the relationship α + β = -(b)/(a) is verified.

The product of the zeroes is 4 × (-2) = -8. Evaluating (c)/(a) gives (-8)/(1) = -8. Since both values are equal, the relationship α β = (c)/(a) is verified.

EXERCISE 2.2 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) $x^2 - 2x - 8$ (ii) $4s^2 - 4s + 1$ (iii) $6x^2 - 3 - 7x$ (iv) $4u^2 + 8u$ (v) $t^2 - 15$ (vi) $3x^2 - x - 4$ 2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) $\frac{1}{4}, -1$ (ii) $\sqrt{2}, \frac{1}{3}$ (iii) $0, \sqrt{5}$ (iv) $1, 1$ (v) $-\frac{1}{4}, \frac{1}{4}$ (vi) $4, 1$

Answer: The zeroes of the polynomial are $4$ and $-2$ . The relationships $\alpha + \beta = -\frac{b}{a} = 2$ and $\alpha\beta = \frac{c}{a} = -8$ are verified.

Step-by-step solution

Step 1: Set the polynomial to zero

To find the zeroes of the quadratic polynomial, we set it equal to 0. Solving this quadratic equation gives the values of $x$ where the polynomial evaluates to 0.

Step 2: Factor the quadratic equation

We split the middle term $-2x$ into $-4x + 2x$ because $(-4)(2) = -8$ and $-4 + 2 = -2$ . Factoring by grouping gives $(x - 4)(x + 2) = 0$ .

Step 3: Determine the zeroes

Setting each linear factor to 0 gives the roots $x = 4$ and $x = -2$ . Thus, the zeroes of the polynomial are $\alpha = 4$ and $\beta = -2$ .

Step 4: Identify coefficients

Comparing $x^2 - 2x - 8$ with the standard quadratic polynomial form $ax^2 + bx + c$ , we get the coefficients $a = 1$ , $b = -2$ , and $c = -8$ .

Step 5: Verify sum of zeroes

The sum of the zeroes is $4 + (-2) = 2$ . Calculating $-\frac{b}{a}$ gives $-\frac{-2}{1} = 2$ . Since both expressions yield 2, the relationship $\alpha + \beta = -\frac{b}{a}$ is verified.

Step 6: Verify product of zeroes

The product of the zeroes is $4 \times (-2) = -8$ . Evaluating $\frac{c}{a}$ gives $\frac{-8}{1} = -8$ . Since both values are equal, the relationship $\alpha \beta = \frac{c}{a}$ is verified.

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