DAV Class 8 Maths Chapter 8 Worksheet 1
Polynomials Worksheet 1
1. Find out whether the given expression is a polynomial or not. If not, give reasons.
(i) \(5x^3 - 4x^2 + \dfrac{1}{2}\)
Answer \(\color{red}{\text{Yes. It is a polynomial}}\). Degree \(=3\).
(ii) \(\sqrt{3}\,z^2 - 5\sqrt{z} + 6\)
Answer \(\color{red}{\text{No, it is not a polynomial}}\). The power of variable 'z' is \( {\dfrac{1}{2}}\) which is not a non-negative integer.
(iii) \(6x^4 + \dfrac{2}{3}x^3 - \dfrac{3}{4}x^2 - 1\)
Answer \(\color{red}{\text{Yes. It is a polynomial}}\). Degree \(=4\).
(iv) \(\dfrac{2}{7}x^3 - 8x^{\frac{3}{2}} + x^2\)
Answer \(\color{red}{\text{No, it is not a polynomial}}\). The power of variable \( x \) is \( {\dfrac{3}{2}}\) which is not a non-negative integer.
(v) \(5x - \dfrac{1}{x} + \dfrac{1}{x^2} - 2\)
Answer \(\color{red}{\text{No, it is not a polynomial}}\). The power of variable \( x \) is \( (-1) \text{ and }(-2) \) which is not a non-negative integer.
(vi) \(p^4 - 3p^3 - p + 1\)
Answer \(\color{red}{\text{Yes. It is a polynomial}}\). Degree \(=4\).
2. Write each of the following polynomials in standard form and also write down their degree.
(i) \(p^6 - 8p^9 + p^7 + 5\)
Solution
\[ \begin{align*} \text{Standard form}&= \color{red} -8p^9 + p^7 + p^6 + 5 \\ \text{Degree}&= \color{red} 9 \end{align*} \]
(ii) \(4z^3 - 3z^5 + 2z^4 + z + 1\)
Solution
\[ \begin{align*} \text{Standard form}&= \color{red} -3z^5 + 2z^4 + 4z^3 + z + 1 \\ \text{Degree}&= \color{red} 5 \end{align*} \]
(iii) \(\left(x + \dfrac{2}{3}\right)\left(x + \dfrac{3}{4}\right)\)
Solution
\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[8pt] \left(x + \frac{2}{3}\right)\left(x + \frac{3}{4}\right) &= x^2 + \left(\frac{2}{3} + \frac{3}{4} \right)x + \left( \frac{\cancel2^1}{\cancel3_1} \times \frac{\cancel3^1}{\cancel4_2}\right) \\[8pt] &= x^2 + \left(\frac{8 + 9}{12} \right)x + \left( \frac{1}{2} \right) \\[8pt] &= x^2 + \frac{17}{12} x + \frac{1}{2} \\ \\ \text{Standard form}&= \color{red} x^2 + \frac{17}{12}x + \frac{1}{2} \\ \text{Degree}&= \color{red} 2 \end{align*} \]
(iv) \(\left(x^2 - \dfrac{2}{3}\right)\left(x^2 + \dfrac{4}{3}\right)\)
Solution
\[ \begin{aligned} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[8pt] \left(x^2-\frac23\right)\left(x^2+\frac43\right) &= (x^2)^2+\left(-\frac23+\frac43\right)x^2+\left(\frac{-2}{3} \times \frac43\right) \\[6pt] &= (x^2)^2+\left(\frac{-2 + 4}{3}\right)x^2+\left(\frac{-8}{9}\right) \\[6pt] &= x^4+\frac23x^2-\frac89 \\[8pt] \text{Standard form}&=\;\color{red}{x^4+\frac23x^2-\frac89} \\ \text{Degree}&=\;\color{red}{4} \end{aligned} \](v) \((z^2 + 5)(z^2 - 6)\)
Solution
\[ \begin{aligned} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[8pt] (z^2+5)(z^2-6) &= (z^2)^2+(5-6)z^2+(5)(-6) \\[6pt] &= z^4+(-1)z^2+(-30) \\[6pt] &= z^4 - z^2 - 30 \\[8pt] \text{Standard form}&=\;\color{red}{z^4 - z^2 - 30} \\ \text{Degree}&=\;\color{red}{4} \end{aligned} \](vi) \((y^3 - 4)(y^3 - 5)\)
Solution
\[ \begin{aligned} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[8pt] (y^3-4)(y^3-5) &= (y^3)^2+(-4-5)y^3+(-4)(-5) \\[6pt] &= y^6 - 9y^3 + 20 \\[8pt] \text{Standard form}&=\;\color{red}{y^6 - 9y^3 + 20} \\ \text{Degree}&=\;\color{red}{6} \end{aligned} \](vii) \((p^2 + 2)(p^2 + 7)\)
Solution
\[ \begin{aligned} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[8pt] (p^2+2)(p^2+7) &= (p^2)^2+(2+7)p^2+(2)(7) \\[6pt] &= p^4 + 9p^2 + 14 \\[8pt] \text{Standard form}&=\;\color{red}{p^4 + 9p^2 + 14} \\ \text{Degree}&=\;\color{red}{4} \end{aligned} \](viii) \(\dfrac{5}{6}z - \dfrac{3}{4}z^2 - \dfrac{2}{3}z^3 + 1\)
Solution
\[ \begin{aligned} \text{Standard form}&=\;\color{red}{-\dfrac{2}{3}z^3 - \dfrac{3}{4}z^2 + \dfrac{5}{6}z + 1} \\ \text{Degree}&=\;\color{red}{3} \end{aligned} \](ix) \(4p + 15p^6 - p^5 + 4p^2 + 3\)
Solution
\[ \begin{aligned} \text{Standard form}&=\;\color{red}{15p^6 - p^5 + 4p^2 + 4p + 3} \\ \text{Degree}&=\;\color{red}{6} \end{aligned} \](x) \(q^{10} + q^6 - q^4 + q^8\)
Solution
\[ \begin{aligned} \text{Standard form}&=\;\color{red}{q^{10} + q^8 + q^6 - q^4} \\ \text{Degree}&=\;\color{red}{10} \end{aligned} \]