Class 9 Practice Level 3 - Polynomials - Set 1

Total questions: 15

If $ \displaystyle x$³ – 6$ \displaystyle x$² + 11$ \displaystyle x$ – 6 is factorized as ($ \displaystyle x-a$)($ \displaystyle x-b$) ($ \displaystyle x-c$), what are the values of $ \displaystyle a$, $ \displaystyle b$, and $ \displaystyle c$?

Topic: Factorization of Cubic Polynomials

$ \displaystyle a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }2,\text{ }c\text{ }=\text{ }3$

$ \displaystyle a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }3,\text{ }c\text{ }=\text{ }2$

$ \displaystyle a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }3,\text{ }c\text{ }=\text{ }1$

$ \displaystyle a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }2,\text{ }c\text{ }=\text{ }1$

None

Using the remainder theorem, find the remainder when 2$ \displaystyle x$³ + 3$ \displaystyle x$² – 5$ \displaystyle x$ + 7 is divided by $ \displaystyle x$ – 2.

Topic: Remainder theorem

None

Factorize the polynomial $ \displaystyle x$³ + 4$ \displaystyle x$² – 3$ \displaystyle x$ – 18 completely given that $ \displaystyle x$ = –3 is a root.

Topic: Factorization Using Synthetic Division

$ \displaystyle (x\text{ }+\text{ }3)({{x}^{2}}+x-6)$

$ \displaystyle (x\text{ }-\text{ }3)({{x}^{2}}+6x-1)$

$ \displaystyle (x\text{ }+2)({{x}^{2}}+2x-9)$

$ \displaystyle (x\text{ }+3)({{x}^{2}}+6x+6)$

None

Given that $ \displaystyle x$ = 1 is a root of the polynomial $ \displaystyle x$³ – 4$ \displaystyle x$² + 5$ \displaystyle x$ – 2, factorize the polynomial completely.

Topic: Factor Theorem

$ \displaystyle \left( {x-1} \right)\left( {x-1} \right)\left( {x+2} \right)$

$ \displaystyle \left( {x-1} \right)\left( {x{}^\text{2}-3x+2} \right)$

$ \displaystyle \left( {x-2} \right)\left( {x+2} \right)\left( {x+1} \right)$

$ \displaystyle \left( {x-1} \right)\left( {x{}^\text{2}-2x+2} \right)$

None

Solve for $ \displaystyle x$ if $ \displaystyle {{x}^{3}}-7x+6=0.$

Topic: Solving Polynomials

$ \displaystyle x$ = 1, 2, –3

$ \displaystyle x$ = –1, –2, 3

$ \displaystyle x$ = 1, –2, 3

$ \displaystyle x$ = –1, 2, –3

None

Using the remainder theorem, determine if $ \displaystyle {x-4}$ is a factor of $ \displaystyle 2{{x}^{3}}-3{{x}^{2}}-11x+12=0.$

Topic: Remainder theorem

Yes

Insufficient information

None of these

None

Divide $ \displaystyle 2{{x}^{3}}+5{{x}^{2}}-4x-7\ \text{by }x+2$ and find the quotient.

Topic: Division of polynomials

$ \displaystyle 2{{x}^{2}}+x-6$

$ \displaystyle 2{{x}^{2}}-x-5$

$ \displaystyle 2{{x}^{2}}+x-5$

$ \displaystyle 2{{x}^{2}}+3x-1$

None

If $ \displaystyle \alpha $ and $ \displaystyle \beta $ are the roots of the quadratic equation $ \displaystyle 2{{x}^{2}}-7x+3=0,$ what is the value of $ \displaystyle {{\alpha }^{2}}+{{\beta }^{2}}$?

Topic: Sum and Product of Roots

$ \displaystyle \dfrac{40}{4}$

$ \displaystyle \dfrac{7}{4}$

$ \displaystyle \dfrac{43}{4}$

$ \displaystyle \dfrac{25}{4}$

None

If $ \displaystyle x$² + 1 = 0, find the roots.

Topic: Complex Numbers and Polynomials

$ \displaystyle x=i,\text{ }x=-i$

$ \displaystyle x=1,\text{ }x=-1$

$ \displaystyle x=i,\text{ }x=1$

$ \displaystyle x=-i,\text{ }x=1$

None

10.

Find the sum and product of the roots of the quadratic equation $ \displaystyle 3{{x}^{2}}-4x+1=0.$

Topic: Sum and Product of Roots

$ \displaystyle \text{Sum }=\dfrac{4}{3},\text{ Product }=\dfrac{1}{3}$

$ \displaystyle \text{Sum }=-\dfrac{4}{3},\text{ Product }=\dfrac{1}{3}$

$ \displaystyle \text{Sum }=\dfrac{4}{3},\text{ Product }=-\dfrac{1}{3}$

$ \displaystyle \text{Sum }=\dfrac{4}{3},\text{ Product }=\dfrac{{-1}}{3}$

None

11.

Divide $ \displaystyle 4{{x}^{3}}+6{{x}^{2}}-2x-3\ \text{by 2}x-1.$ What is the quotient?

Topic: Polynomial long Division

$ \displaystyle 2{{x}^{2}}+5x+1$

$ \displaystyle 2{{x}^{2}}+4x+1$

$ \displaystyle 2{{x}^{2}}+3x+1$

$ \displaystyle 2{{x}^{2}}+3x-2$

None

12.

If the discriminant of the quadratic equation $ \displaystyle 4{{x}^{2}}-12x+k=0$ is zero, find the value of $ \displaystyle k$.

Topic: Nature of Roots

$ \displaystyle k=9$

$ \displaystyle k=6$

$ \displaystyle k=12$

$ \displaystyle k=18$

None

13.

What is the degree of the polynomial $ \displaystyle p\left( x \right)=\left( {{{x}^{2}}+1} \right)\left( {{{x}^{3}}-x} \right)?$

Topic: Degree of Polynomials

None

14.

Using the Factor Theorem, determine if $ \displaystyle {x-2}$ is a factor of $ \displaystyle {{x}^{3}}-4{{x}^{2}}-5x+10.$

Topic: </strongFactor Theorem

Yes

No

Insufficient information

None of these
None

15.
Find one rational root of the cubic equation $ \displaystyle {{x}^{3}}-3{{x}^{2}}-4x+12=0$ using the trial- and-error method.

Topic: Roots of a Cubic Equation

$ \displaystyle x=3$

$ \displaystyle x=2$

$ \displaystyle x=-3$

$ \displaystyle x=-2$
None

1 out of 15

Time's up

Class 9 Practice Level 3 - Polynomials - Set 1

Welcome back