How do you find the quadratic polynomial when sum of zeros and product of zeros are given?
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Solution Let the quadratic polynomial be ax^2+bx+c and it's zero be alpha and beta Then α+β=3 α*β=2 we know that α+β= -b/a =3/1 α*β=c/a =2/1 From this a=1 b=-3 c=2 Therefore the required polynomial =1*x^2+(-3x)+2 =x^2-3x+2Roots of a PolynomialA "root" (or "zero") is where the polynomial is equal to zero: Put simply: a root is the x-value where the y-value equals zero. General PolynomialIf we have a general polynomial like this: f(x) = axn + bxn-1 + cxn-2 + ... + z Then:
Which can sometimes help us solve things. How does this magic work? Let's find out ... FactorsWe can take a polynomial, such as: f(x) = axn + bxn-1 + cxn-2 + ... + z And then factor it like this: f(x) = a(x−p)(x−q)(x−r)... Then p, q, r, etc are the roots (where the polynomial equals zero) QuadraticLet's try this with a Quadratic (where the variable's biggest exponent is 2): ax2 + bx + c When the roots are p and q, the same quadratic becomes: a(x−p)(x−q) Is there a relationship between a,b,c and p,q? Let's expand a(x−p)(x−q): a(x−p)(x−q) Now let us compare:
We can now see that −a(p+q)x = bx, so: −a(p+q) = b p+q = −b/a And apq = c, so: pq = c/a And we get this result:
This can help us answer questions. Example: What is an equation whose roots are 5 + √2 and 5 − √2The sum of the roots is (5 + √2) + (5 − √2) =
10 And we want an equation like: ax2 + bx + c = 0 When a=1 we can work out that:
Which gives us this result x2 − (sum of the roots)x + (product of the roots) = 0 The sum of the roots is 10, and product of the roots is 23, so we get: x2 − 10x + 23 = 0 And here is its plot: (Question: what happens if we choose a=−1 ?) CubicNow let us look at a Cubic (one degree higher than Quadratic): ax3 + bx2 + cx + d As with the Quadratic, let us expand the factors: a(x−p)(x−q)(x−r) And we get:
We can now see that −a(p+q+r)x2 = bx2, so: −a(p+q+r) = b p+q+r = −b/a And −apqr = d, so: pqr = −d/a This is interesting ... we get the same sort of thing:
(We also get pq+pr+qr = c/a, which can itself be useful.) Higher PolynomialsThe same pattern continues with higher polynomials. In General:
Sum of zeroes = α + β =√2 Product of zeroes = α β = 1/3 ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:- x2–(α+β)x +αβ = 0 x2 –(√2)x + (1/3) = 0 3x2-3√2x+1 = 0 Thus, 3x2-3√2x+1 is the quadratic polynomial. How do you find a quadratic polynomial when its zeros are given?How to Find a Quadratic Polynomial if Zeros are Given?. Find the sum.. Find the product.. Substitute the values in the expression x2 - (sum of zeros)x + (product of the zeros) to get the required quadratic polynomial.. What is the quadratic polynomial whose sum and the product of zeros?- (sum of zeros) x + (product of zeros)=0. Complete step-by-step answer: A quadratic polynomial is a polynomial of degree 2 or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
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