Question 1

Define a polynomial and give an example of a linear polynomial in one variable.

Accepted Answer

A polynomial is an algebraic expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. The general form of a polynomial in one variable x is $p(x) = an x^n + a{n-1} x^{n-1} +

Question 2

Write the quadratic polynomial whose sum and product of zeroes are $-3$ and $2$ respectively.

Accepted Answer

If the sum of zeroes $(\alpha + \beta)$ and the product of zeroes $(\alpha \beta)$ of a quadratic polynomial are given, the general quadratic polynomial can be expressed using the formula $P(x) = kx^2 - (	ext{sum of zeroes})x + (	ext{product of zeroes})$, where $k$ is a non-zero real constant. Sub

Question 3

Find the zeroes of the polynomial $p(x) = x^2 - 3$ and verify the relationship.

Accepted Answer

To find the zeroes of $p(x) = x^2 - 3$, we set $p(x) = 0$, which gives $x^2 - 3 = 0$. We can rewrite this using the difference of squares identity as $(x - \sqrt{3})(x + \sqrt{3}) = 0$. This yields two possible solutions: $x = \sqrt{3}$ and $x = -\sqrt{3}$. Thus, the zeroes of the given quadratic po

Question 4

Find the value of the polynomial $5x - 4x^2 + 3$ at $x = -1$.

Accepted Answer

Let the given polynomial be represented as $p(x) = 5x - 4x^2 + 3$. To find its value at $x = -1$, we substitute $x = -1$ wherever $x$ appears in the polynomial expression. This gives $p(-1) = 5(-1) - 4(-1)^2 + 3$. Now, evaluating each term step by step: $5(-1) = -5$, $(-1)^2 = 1$ so $-4(1) = -4$, an

Question 5

Can a quadratic polynomial have no zeroes? Explain with a graphical reference.

Accepted Answer

Yes, a quadratic polynomial can have no real zeroes. Geometrically, the graph of a quadratic polynomial $y = ax^2 + bx + c$ is a parabola that either opens upwards or downwards. If the parabola lies entirely above the x-axis or entirely below the x-axis, it never intersects or touches the x-axis at

Polynomials