Question

Find the zeros of a polynomial \[{v^2} + 4\sqrt 3 v - 15 = 0\].

Answer 1

Hint: The polynomial is quadratic in degree two and has two roots. The roots of the polynomial \[a{x^2} + bx + c = 0\] are given by \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].

Complete step-by-step answer:
Polynomials are expressions that are composed of algebraic terms. A polynomial is composed of constants, variables, and exponents. Examples of polynomials are \[5x + 6\], \[{x^2} + 6\], \[{x^{100}}\] and so on.

The degree of the polynomial is defined as the highest power to which the variable is raised in the expression. The degree of \[{x^{100}} - 1\] is 100.

A quadratic polynomial is a polynomial with a degree or highest power equal to two. An example of a quadratic polynomial is \[{x^2} + 6\].

The standard form of a quadratic polynomial is \[a{x^2} + bx + c\].

Zeroes of the polynomial are defined as the values of the variable at which the value of the polynomial becomes zero.

Any polynomial has as many roots equal to the degree of the polynomial. Hence, the quadratic polynomial has two roots. These two roots may be equal or unequal depending on the coefficients of the terms in the expression.

For the quadratic polynomial in its standard form, the zeroes of the polynomial is given by the formula as follows:

\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}...........(1)\]

The roots of the equation \[{v^2} + 4\sqrt 3 v - 15 = 0\] are determined as follows:

\[v = \dfrac{{ - 4\sqrt 3 \pm \sqrt {{{(4\sqrt 3 )}^2} - 4(1)( - 15)} }}{{2(1)}}\]

\[v = \dfrac{{ - 4\sqrt 3 \pm \sqrt {48 + 60} }}{2}\]

\[v = \dfrac{{ - 4\sqrt 3 \pm \sqrt {108} }}{2}\]

The value of \[\sqrt {108} \] is \[6\sqrt 3 \]. Hence, we have:

\[v = \dfrac{{ - 4\sqrt 3 \pm 6\sqrt 3 }}{2}\]

\[v = - 2\sqrt 3 \pm 3\sqrt 3 \]

\[v = \sqrt 3 ;v = - 5\sqrt 3 \]

Hence, the zeroes are \[\sqrt 3 \] and \[ - 5\sqrt 3 \].

Note: You can also split \[4\sqrt 3 \] into \[5\sqrt 3 \] and \[ - \sqrt 3 \] and proceed by taking common terms to find the zeros of the given quadratic equation.