1. The graph of a quadratic polynomial $p(x)$ passes through the points $(-6,0),(0,-30)$, $(4,-20)$ and $(6,0)$. The zeroes of the polynomial are:
(a) $-6,0$
(b) 4,6
(c) $-30,-20$
(d) $-6,6$
[CBSE SQP 2024]
Ans. (d) $-6,6$
Explanation: The quadratic polynomial passes through the points $(-6,0)$ and $(6,0)$, so its zeroes are $x=-6$ and $x=6$.
Concept
The roots (or zeroes) of a quadratic polynomial $p(x)$ are the values where graph of the polynomial intersects or touches the x-axis.
2. What should be added from the polynomial $x^2-5 x+4$, so that 3 is the zero of the resulting polynomial?
(a) 1
(b) 2
(c) 4
(d) 5
[CBSE 2024]
Ans. (b) 2
Explanation:
Let,
$$
f(x)=x^2-5 x+4
$$
Let $p$ should be added to $f(x)$ then 3 becomes zero of polynomial.
So,
$$
f(3)+p=0
$$
$\Rightarrow \quad 3^2-5 \times 3+4+p=0$
$\Rightarrow \quad 9+4-15+p=0$
$\Rightarrow \quad-2+p=0$
$\Rightarrow$
$$
p=2
$$
So, 2 should be added.
Concept Applied
$\rightarrow$ If any number is a zero of polynomial, then value of the polynomial becomes zero at that number.
3. A quadratic polynomial, the product and sum of whose zeroes are 5 and 8 , respectively is:
(a) $k\left[x^2-8 x+5\right]$
(b) $k\left[x^2+8 x+5\right]$
(c) $k\left[x^2-5 x+8\right]$
(d) $k\left[x^2+5 x+8\right]$
[CBSE Term-1 Std. 2021]
Ans. (a) $k\left[x^2-8 x+5\right]$
Explanation: We know that, a quadratic polynomial with sum ( S ) and product ( P ) of zeroes is given as:
i.e.,
$$
\begin{aligned}
& k\left[x^2-S x+P\right] \\
& k\left[x^2-8 x+5\right]
\end{aligned}
$$
Concept
The quadratic polynomial is: $x^2$-(sum of zeroes) $x+$ (product of zeroes)
