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Polynomials – Sum And Product Of The Roots

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  • 1. Polynomials – Sum and Product of the Roots
  • 2. Do Now: f(x)=(2x – 3)(x – 2)(x + 4) a. Write f(x) in standard a. 2x3 + x2 - 22x + 24 form b. Find the roots c. - 4, 2, 3/2 c. Find the sum of the roots c. - ½ d. Find the product of the d. - 12 roots e. ½ e. Write the ratio f. 12 f. Write the ratio
  • 3. Formulas Sum of the roots of a Product of the roots of a polynomial: polynomial: b c   1  n a a Where b is the coefficient of Where c is the constant and the term one degree less n is the degree of the than first term and a is the polynomial. leading coefficient
  • 4. Ex. 1 Find the sum and product of the roots of f(x) = 3x6 – 4x4 + 3x3 –x + 5 Sum = - b/a Product = c/a (even degree) b = 0 (no x5 term) c=5 a=3 a=3 Sum = 0/3=0 Product = 5/3
  • 5. Ex. 2 Write an equation of a polynomial whose roots are 2 + i, 3 - √5, and 4 Steps: 2+i, 2-i Sum = 4, Product = 5 2. Write conjugate roots Sum = -b/a. Product = c/a 3. Rewrite irrational and If a = 1, b = -4 C= 5 Factor is x2 – 4x + 5 complex roots as quadratic factors using 3 - √5, 3 + √5 sum and product Sum = 6 Product = 4 Sum = - b/a product = c/a 4. Write f(x) with all factors If a = 1, b = - 6 c=4 Factor is x2 – 6x + 4 Factor x – 4 f(x) = (x-4)(x2 – 4x + 5)(x2 – 6x + 4)
  • 6. Ex. 3 If f(x) = 5x4 +px3 +q has roots 2 + i and 5 – i, find p and q. 2+i,2 – i, 5 – i, 5 + i Steps: Sum: 4 10 14 2. Write conjugate roots Product: 5 26 130 3. Find sum and product of all roots Sum = -b/a 4. Use a from f(x) and find “b” 14 = -b/5 and “c” using sum and b = -70 = p product formulas Product = c/a (4th degree) 130 = c/5 c = 650 = q