Question

What is the relationship between the roots and the coefficients of a polynomial?

Answer 1

Hint: A polynomial is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a positive integral power (like\[a + bx + c{x^2}\])

Complete step-by-step solution:
Let \[{\alpha _1},{\alpha _2},{\alpha _3}...{\alpha _n}\] are the roots of any polynomial which is given by an expression,
\[f(x) = {a_0}{x^n} + {a_1}{x^{n - `}} + {a_2}{x^{n - 2}} + ... + {a_{n - 1}}x + {a_n} = 0\]
Therefore, we can write
\[f(x) = {a_0}(x - {\alpha _1})(x - {\alpha _2})(x - {\alpha _3})...(x - {\alpha _n})\]
Now, equating the right hand side of both the above equation, we get
\[{a_0}(x - {\alpha _1})(x - {\alpha _2})(x - {\alpha _3})...(x - {\alpha _n}) = {a_0}{x^n} + {a_1}{x^{n - `}} + {a_2}{x^{n - 2}} + ... + {a_{n - 1}}x + {a_n} = 0\]
Now, comparing coefficient of \[{x^{n - 1}}\] on both sides we get
\[{s_1} = {\alpha _1} + {\alpha _2} + {\alpha _3} + ... + {\alpha _n} = \sum {{\alpha _i}} = - \dfrac{{{a_1}}}{{{a_0}}}\]
\[\Rightarrow {s_1} = - \dfrac{{\text{coeff. of}({x^{n - 1}})}}{{\text{coeff. of} ({x^n})}}\]
Comparing coefficient of \[{x^{n - 2}}\] on both sides, we get
\[{s_1} = {\alpha _1}{\alpha _2} + {\alpha _1}{\alpha _3} + ... = \sum\limits_{i \ne j} {{\alpha _i}{\alpha _j}} = {( - 1)^2}\dfrac{{{a_2}}}{{{a_0}}}\]
\[\Rightarrow {s_1} = {( - 1)^2}\dfrac{{\text{coeff. of}({x^{n - 2}})}}{{\text{coeff. of}({x^n})}}\]
Particular case: Quadratic equation \[a{x^2} + bx + c = 0\],
$>$The solutions of the quadratic equation, \[a{x^2} + bx + c = 0\] is given by \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
$>$The expression \[{b^2} - 4ac = D\] is called the discriminant of the quadratic equation.
$>$If α & β are the roots of the quadratic equation \[a{x^2} + bx + c = 0\] then;
1) \[\alpha + \beta = - \dfrac{b}{a}\]
2) \[\alpha \beta = \dfrac{c}{a}\]
3) \[\left| {\alpha - \beta } \right| = \dfrac{{\sqrt D }}{{\left| a \right|}}\]
$>$Quadratic equation whose roots are α & β is \[(x - \alpha )(x - \beta ) = 0\]i.e.
\[{x^2} - (\alpha + \beta )x + \alpha \beta = 0\] i.e. \[{x^2} - \](sum of roots) \[x\]\[ + \]product of roots\[ = 0\]
Nature of Roots: Consider the quadratic equation \[a{x^2} + bx + c = 0\] where \[a,b,c \in R\& a \ne 0\] then;
$\to$ \[D > 0 \Leftrightarrow \] Roots are real & distinct (unequal).
$\to$ \[D = 0 \Leftrightarrow \] Roots are real & coincident (equal).
$\to$ \[D < 0 \Leftrightarrow \] Roots are imaginary.

Note:
$>$A polynomial is a combination of terms that are only added, subtracted or multiplied.
$>$A quadratic polynomial with real coefficients is of the form \[a{x^2} + bx + c\], where a, b, c are real numbers with \[a \ne 0\] .
$>$The highest exponent of the variable in the equation is called the degree of polynomial.
$>$Polynomials of degrees \[1,2and3\] are called linear, quadratic and cubic polynomials respectively.