For a quadratic polynomial ax² + bx + c, the sum of the zeroes (roots) is given by -(b)/(a) and the product of the zeroes is given by (c)/(a). In our polynomial 3x² - 13x - 10, we have a=3, b=-13, and c=-10.

For a quadratic polynomial ax² + bx + c, the sum of the zeroes (roots) is given by -(b)/(a) and the product of the zeroes is given by (c)/(a). In our polynomial 3x² - 13x - 10, we have a=3, b=-13, and c=-10.

Substitute the values of a, b, and c into the formulas for the sum and product of the zeroes. The sum α + β becomes -(-13)/(3) = (13)/(3), and the product αβ becomes (-10)/(3).

Expand the given expression (3α + 1)(3β + 1) using the distributive property (FOIL method). This yields 9αβ + 3α + 3β + 1.

Notice that 3α + 3β can be factored as 3(α + β). This simplifies the expression to 9αβ + 3(α + β) + 1.

If $\alpha$ , $\beta$ are the zeroes of the polynomial $3x^2 - 13x - 10$ , then find the value of $(3\alpha + 1)(3\beta + 1)$ .

Answer: -16

Step-by-step solution

Step 1: Identify coefficients and sum/product of roots

For a quadratic polynomial $ax^2 + bx + c$ , the sum of the zeroes (roots) is given by $-\frac{b}{a}$ and the product of the zeroes is given by $\frac{c}{a}$ . In our polynomial $3x^2 - 13x - 10$ , we have $a=3$ , $b=-13$ , and $c=-10$ .

Step 2: Calculate sum and product of zeroes

Substitute the values of $a$ , $b$ , and $c$ into the formulas for the sum and product of the zeroes. The sum $\alpha + \beta$ becomes $-\frac{-13}{3} = \frac{13}{3}$ , and the product $\alpha\beta$ becomes $\frac{-10}{3}$ .

Step 3: Expand the expression

Expand the given expression $(3\alpha + 1)(3\beta + 1)$ using the distributive property (FOIL method). This yields $9\alpha\beta + 3\alpha + 3\beta + 1$ .

Step 4: Factor out common term

Notice that $3\alpha + 3\beta$ can be factored as $3(\alpha + \beta)$ . This simplifies the expression to $9\alpha\beta + 3(\alpha + \beta) + 1$ .

Step 5: Substitute values and calculate

Substitute the calculated values of $\alpha\beta = -\frac{10}{3}$ and $\alpha + \beta = \frac{13}{3}$ into the simplified expression. Perform the multiplication and addition to find the final value.

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If α\alphaα, β\betaβ are the zeroes of the polynomial 3x2−13x−103x^2 - 13x - 103x2−13x−10, then find the value of (3α+1)(3β+1)(3\alpha + 1)(3\beta + 1)(3α+1)(3β+1).