If α, β are the zeroes of the polynomial 3x2−13x−10, then find the value of (3α+1)(3β+1).
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Step-by-Step Solution
Step 1: Identify coefficients and sum/product of roots
For a quadratic polynomial ax2+bx+c, the sum of the zeroes (roots) is given by −ab and the product of the zeroes is given by ac. In our polynomial 3x2−13x−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 α+β becomes −3−13=313, and the product αβ becomes 3−10.
Step 3: Expand the expression
Expand the given expression (3α+1)(3β+1) using the distributive property (FOIL method). This yields 9αβ+3α+3β+1.
Step 4: Factor out common term
Notice that 3α+3β can be factored as 3(α+β). This simplifies the expression to 9αβ+3(α+β)+1.
Step 5: Substitute values and calculate
Substitute the calculated values of αβ=−310 and α+β=313 into the simplified expression. Perform the multiplication and addition to find the final value.