# Ex.2.3 Q5 Polynomials Solution - NCERT Maths Class 10

## 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 \(r(x) =\,0\)

## Text Solution

**What is known?**

(i) deg \(p(x)=\) deg \(q(x)\)

(ii) deg \(q(x)=\) deg \(r(x)\)

(iii) deg \(r(x) =\,0\)

**What is unknown?**

Examples of polynomials \(p(x),\, g(x), \,q(x)\) and \(r(x),\) which satisfy the division algorithm

**Reasoning:**

To solve this question, follow some steps

In case (i), assume polynomial *\(p(x)\)* whose degree is equal to degree of \(q(x),\) then put the values of \(p(x), \,g(x), \,q(x) \) and *\(r(x)\)*

In the division algorithm, if L.H.S is equal to R.H.S, then the division algorithm is satisfied.

In case (ii), assume polynomial *\(p(x)\)* in which degree of quotient *\(q(x)\)* is equal to the degree of \(r(x),\) then put the values of \(p(x), \,g(x),\, q(x)\) and *\(r(x)\)* in the division algorithm. If L.H.S is equal to R.H.S then the division algorithm is satisfied.

In case (iii), assume polynomial *\(p(x)\)* in which degree of remainder *\(r(x)\)* is equal to zero, then put the values of \(p(x),\, g(x),\, q(x)\) and *\(r(x)\)* in the division algorithm.

If L.H.S is equal to R.H.S, then the division algorithm is satisfied.

Use the below given statement of Division algorithm to solve this question

Division algorithm

Dividend \(=\) Divisor \(\times\) Quotient \(+\) Remainder

According to division algorithm, if *\(p(x)\)* and *\(g(x)\)* are two polynomials with \(g\left( x \right) \ne 0,\) then we can find polynomial *\(q(x)\)* and *\(r(x)\)* such that

\[p\left( x \right)=g\left( x \right)\times q\left( x \right)+r\left( x \right)\]

Where \(r\left( x \right) = 0\) or degree of \(r(x) < \) degree of *\(g(x)\)*

Degree of polynomial is the highest power of the variable in the polynomial.

Put the given values in the above equation and simplify it, get the value of \(g(x).\)

**Steps:**

(i) deg \(p(x)=\) deg \(q(x)\)

Degree of quotient will be equal to the degree of dividend when divisor is constant (i.e. when any polynomial is divided by a constant).

Let us assume the division of \(6{x^2} + {\text{ }}2x{\text{ }} + 2\) by \(2\)

\[\begin{align} p\left( x \right)&=6{{x}^{2}}+2x\text{ }+2 \\ g\left( x \right)&=2 \\q\left( x \right)&=3{{x}^{2}}+\text{ }x+1,\; r\left( x \right)\text{ }=\text{ }0 \\ \end{align}\]

Degree of *\(p(x)\)* and *\(q(x)\)* is same i.e. \(2.\)

Checking for division algorithm:

\[\begin{align} p(x) &=g(x) \times q(x)+r(x) \\ 6 x^{2}+2 x+2 &=2\left(3 x^{2}+x+1\right)+0 \\ &=6 x^{2}+2 x+2 \end{align}\]

Thus, the division algorithm is satisfied.

(ii) deg \(q(x)=\) deg \(r(x)\)

Let us assume the division of \({x^3} + x\) by \({x^2}\)

\[\begin{align}p\left( x \right)&={{x}^{3}}+\text{ }x \\ g\left( x \right)&={{x}^{2}} \\ q\left( x \right)\,&=x,\;r\left( x \right)\text{ }=x \\\end{align}\]

Clearly, degree of *\(p(x)\)* and *\(q(x)\)* is same i.e. \(1.\)

Checking for division algorithm

\[\begin{align}p\left( x \right)&=g\left( x \right)\times q\left( x \right)+r\left( x \right) \\{{x}^{3}}+x&=({{x}^{2}}\times x)+x \\{{x}^{3}}+x &= {{x}^{3}}+x \end{align}\]

Thus, the division algorithm is satisfied.

(iii) deg \(r(x) =0\)

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of \({x^3} + 1\) by \({x^2}.\)

\(p\left( x \right)={{x}^{3}}+1,\; \quad g\left( x \right)={{x}^{2}},\\ q\left( x \right)=\text{ }x \;\; \text{and} \;\; r\left( x \right)=1\)

Clearly, the degree of *\(r(x)\)* is \(0.\)

Checking for division algorithm

\[\begin{align}p\left( x \right)&=g\left( x \right)\times q\left( x \right)+r\left( x \right) \\ {{x}^{3}}+1&=\text{ }({{x}^{2}}\times x)+1 \\ {{x}^{3}}+1&={{x}^{3}}+1 \\\end{align}\]

Thus, the division algorithm is satisfied.