Class 9 Practice Level 3 – Polynomials – Set 1 Leave a Comment / By Gulam Hamza / July 18, 2024 Class 9 Practice Level 3 - Polynomials - Set 1 Total questions: 15 1. 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 2. 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 1 3 5 7 None 3. 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 4. 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 5. 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 6. 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 No Insufficient information None of these None 7. 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 8. 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 9. 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 5 4 3 2 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
Sign in with Google
Sign in with Facebook