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CBSE - Class 10 Mathematics Polynomials Worksheet
One of the factors of (25x2 - 1) + (1 + 5x)2 is____
If the sum of the zeroes of the polynomial is 6, then the value of k is
2
b.4
c.-2
d.-4
If zeros of the quadratic polynomial ,
are equal , then
c and a have opposite signs
b.c and b have opposite signs
c.c and a have the same sign
d.c and b have the same sign
The degree of the polynomial is
2
b.3
c.4
d.1
If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is
2
b.-2
c.4
d.-4
1
b.2
c.4
d.5
If α and β are the zeros of the polynomial , then a polynomial having
is its zeros is
The factorisation of 4x2 + 8x + 3 is _______
What is the number(s) of zeores that a cubic polynomial has/have
0
b.1
c.2
d.3
Obtain the zeroes of the quadratic polynomial: pqx2 + (q2-Pr)x- qr?
Find the zeroes of the quadratic polynomial: x2 + 19x + 90?
if α and β are the given zeros of the polynomial f(x)= such that α-β=1, find the value of k.
6
b.8
If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
3,0
b.4,1
c.3,1
d.4,0
The number of zeros for a polynomial P(x) where grph of y=f(x) is given in figure is
2
b.3
c.4
d.1
What is an example of a 4th degree polynomial with exactly 4 terms?
Find the zeroes of the x2 – 2x – 8?
Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
c/a
b.-c/a
b/a
d.-b/a
solve
1/3
b.-1/3
c.2/3
d.-2/3
A quadratic polynomial, whose zeroes are -3 and 4, is
x²- x + 12
b.x² + x + 12
c.2x² + 2x – 24
Worksheet Answers
10x
(2x+1)(2x+3)
- q/p, r/q
- 9, - 10
An example is -x4 - x3 + 3x + 2.
4, -2
Solution:
Given: A quadratic polynomial $p(u) = 4u^2 + 8u$.
To Find: The zeroes of the polynomial and verify the relationship between the zeroes and the coefficients of the polynomial.
Step 1: Finding the zeroes of the polynomial
To find the zeroes of the polynomial $p(u)$, we set $p(u) = 0$.
$4u^2 + 8u = 0$
We factor out the greatest common factor, which is $4u$:
$4u(u + 2) = 0$ [Using the distributive property of multiplication over addition]
For the product to be zero, either $4u = 0$ or $u + 2 = 0$ [Zero Product Property].
Case 1: $4u = 0 \implies u = 0$
Case 2: $u + 2 = 0 \implies u = -2$
Thus, the zeroes of the polynomial are $\alpha = 0$ and $\beta = -2$.
Step 2: Identifying coefficients
Comparing the given polynomial $4u^2 + 8u$ with the standard form $au^2 + bu + c$, we have:
$a = 4$
$b = 8$
$c = 0$
Step 3: Verifying the relationship between zeroes and coefficients
The relationships to verify are:
1. Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
2. Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$
Verification of Sum of Zeroes:
Sum of zeroes = $\alpha + \beta = 0 + (-2) = -2$
$-\frac{b}{a} = -\frac{8}{4} = -2$
Since $-2 = -2$, the relationship $\alpha + \beta = -\frac{b}{a}$ is verified.
Verification of Product of Zeroes:
Product of zeroes = $\alpha \cdot \beta = 0 \cdot (-2) = 0$
$\frac{c}{a} = \frac{0}{4} = 0$
Since $0 = 0$, the relationship $\alpha \cdot \beta = \frac{c}{a}$ is verified.
Final Answer: The zeroes of the polynomial $4u^2 + 8u$ are $0$ and $-2$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $-2$ and the product of zeroes is $0$.
