Give examples of polynomials p(x),
[question] Question. Give examples of polynomials p(x), g(x),q(x) and r(x), which satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0. [/question] [solution] Solution: (i) $p(x)=2 x^{2}+2 x+8, g(x)=2 x^{0}=2$ $q(x)=x^{2}+x+4 ; r(x)=0$ (ii) $\mathrm{p}(\mathrm{x})=2 \mathrm{x}^{2}+2 \mathrm{x}+8 ; \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{x}+9$; $q(x)=2 ; r(x)=-10$ (iii) $\mathrm{p}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}+5 ; \mathrm{g}(\mathrm...
On dividing $x^{3}-3 x^{2}+x+2$ by a polynomial $g(x)$
[question] Question. On dividing $x^{3}-3 x^{2}+x+2$ by a polynomial $g(x)$, the quotient and remainder were $x$ $-2$ and $-2 x+4$, respectively. Find $g(x)$ [/question] [solution] Solution: $\left(x^{3}-3 x^{2}+x+2\right)=g(x) \times(x-2)+(-2 x+4)$ $\Rightarrow x^{3}-3 x^{2}+x+2+2 x+-4=g(x) \times(x-2)$ $\Rightarrow x^{3}-3 x^{2}+3 x-2=g(x) \times(x-2)$ $g(x)=\frac{x^{3}-3 x^{2}+3 x-2}{x-2}$ So, $g(x)=x^{2}-x+1$ [/solution]...
Obtain all other zeros of $3 x^{4}+6 x^{3}-2 x^{2}-10 x-5$
[question] Question. Obtain all other zeros of $3 x^{4}+6 x^{3}-2 x^{2}-10 x-5$, if two of its zeros are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$ [/question] [solution] Solution: Two of the zeros of $3 x^{4}+6 x^{3}-2 x^{2}-10 x-5$ are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$. $\Rightarrow\left(x-\sqrt{\frac{5}{3}}\right)\left(x+\sqrt{\frac{5}{3}}\right)$ is a factor of the polynomial. i.e., $\mathrm{x}^{2}-\frac{\mathbf{5}}{\mathbf{3}}$ is a factor. i.e., $\left(3 x^{2}-5\right)$ is...
Check whether the first polynomial is a factor of the second polynomial
[question] Question. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial. (i) $t^{2}-3,2 t^{4}+3 t^{3}-2 t^{2}-9 t-12$ (ii) $x^{2}+3 x+1,3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ (iii) $x^{3}-3 x+1, x^{5}-4 x^{3}+x^{2}+3 x+1$ [/question] [solution] Solution: Hence, $t^{2}-3$ is a factor of $2 t^{4}+3 t^{3}-2 t^{2}-9 t-12$ (ii) $x^{2}+3 x+1$ Hence, $x^{2}+3 x+1$ is a factor of $3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ (iii) $x^{3}-3 x+1$ He...
Divide the polynomial p(x) by the polynomial
[question] Question. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following : (i) $\mathrm{p}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}+5 \mathrm{x}-3, \mathrm{~g}(\mathrm{x})=\mathrm{x}^{2}-2$ (ii) $p(x)=x^{4}-3 x^{2}+4 x+5, g(x)=x^{2}+1-x$ (iii) $\mathrm{p}(\mathrm{x})=\mathrm{x}^{4}-5 \mathrm{x}+6, \mathrm{~g}(\mathrm{x})=2-\mathrm{x}^{2}$ [/question] [solution] Solution: Hence, Quotient $q(x)=x-3$ and Remainder $r(x)=7 x-9$ (ii) $x^{2...
Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively
[question] Question. Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively (i) $\frac{1}{4},-1$ (ii) $\sqrt{2}, \frac{1}{3}$ (iii) $\mathbf{0}, \sqrt{\mathbf{5}}$ (iv) 1,1 $(V)-\frac{1}{4}, \frac{1}{4}$ (vi) 4,1 [/question] [solution] Solution: (i) Required polynomial = $x^{2}-($ sum of zeros $) x+$ product of zeros $=x^{2}-\frac{1}{4} x-1$ $=\frac{1}{4}\left(4 x^{2}-x-1\right)$ (ii) Required polynomial = $x^{2}-($ sum of zeros $) x+$ product of...
Find the zeros of the following quadratic polynomials and verify
[question] Question. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients. (i) $x^{2}-2 x-8$ (ii) $4 \mathrm{~s}^{2}-4 \mathrm{~s}+1$ (iii) $6 x^{2}-3-7 x$ (iv) $4 u^{2}+8 u$ (v) $t^{2}-15$ (vi) $3 x^{2}-x-4$ [/question] [solution] Solution: (i) $x^{2}-2 x-8=x^{2}-4 x+2 x-8$ $=x(x-4)+2(x-4)=(x+2)(x-4)$ Zeroes are – 2 and 4. Sum of the zeros $=(-2)+(4)=2=\frac{-(-2)}{1}=\frac{-(\text { Coefficient of } \mathbf{x})}{\left(\text { ...