SOLUTION: Find an equation for f(x), the polynomial of smallest degree with real coefficients such that f(x) bounces off of the x-axis at −5, breaks through the x-axis at −2, has

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find an equation for f(x), the polynomial of smallest degree with real coefficients such that f(x) bounces off of the x-axis at −5, breaks through the x-axis at −2, has      Log On


   

Question 1011139: Find an equation for f(x), the polynomial of smallest degree with real coefficients such that f(x) bounces off of the x-axis at −5, breaks through the x-axis at −2, has complex roots of −2+2i and −4−2i and passes through the point (0,−54).
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Real roots are -5 and -2;
Other complex roots are -2+2i, -2-2i, -4-2i, -4+2i.
Further, f%28x%29=a%2A%28polynomialDegree6%29 and you use the y-intercept to finally find a.

Start with .

a%28x%2B5%29%28x%2B2%29%28x%2B2-2i%29%28x%2B2%2B2i%29%28x%2B4%2B2i%29%28x%2B4-2i%29

a%28x%2B5%29%28x%2B2%29%28%28x%2B2%29%5E2%2B4%29%28%28x%2B4%29%5E2%2B4%29----this really combined two steps including knowing about difference of squares and of i*i=-1;

a%28x%2B5%29%28x%2B2%29%28x%5E2%2B4x%2B4%2B4%29%28x%5E2%2B8x%2B16%2B4%29

a%28x%2B5%29%28x%2B2%29%28x%5E2%2B4x%2B8%29%28x%5E2%2B8x%2B20%29--------at this stage, the function is in purely factored form without showing complex nor imaginary expressions - only real coefficients. You CAN still use this to find the value for a, making use of the given y-intercept; and you may still want to continue the full multiplication into general form.