Question

Using a long division method check whether the second polynomial is the factor of the first polynomial.
\[4{{q}^{3}}-6{{q}^{2}}-4q+3;2q-1\]

Answer 1

Hint: In order to find if \[2q-1\] is a factor \[4{{q}^{3}}-6{{q}^{2}}-4q+3\] using long division method, we must perform division process by considering the dividend as \[4{{q}^{3}}-6{{q}^{2}}-4q+3\] and the divisor as \[2q-1\]. After performing the division, if we obtain the remainder as zero, then we can conclude that the second polynomial is the factor of the first polynomial.

Complete step-by-step solution:
Now let us learn about the process of long division upon the polynomials.
1.We have to divide the first term of numerator by the first term of denominator i.e. divisor and the obtained answer should be placed in the place of quotient.
2.Now we should multiply the next term of the divisor with the first term written in the quotient and write the obtained answer in the second term’s place of the dividend.
3.Next, we have to subtract the polynomials and write the difference between them.
4. We have to follow the same process again and again until we obtain the remainder zero or the polynomial such that the degree of the divisor is greater than the degree of the remainder.
Now let us perform the long division upon the given polynomials.
\[\begin{align}
& \text{2q-1}\overset{\text{2}{{\text{q}}^{\text{2}}}\text{- 2q-
3}}{\overline{\left){\text{4}{{\text{q}}^{\text{3}}}\text{-6}{{\text{q}}^{\text{2}}}\text{-4q+3}}\right.}}
\\
& \,\,\,\,\,\,\,\,\,\,\,\,\underline{\text{ -4}{{\text{q}}^{\text{3}}}\text{-2}{{\text{q}}^{\text{2}}} \downarrow} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\text{ -4}{{\text{q}}^{\text{2}}}\text{- 4q} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\underline{\text{ -4}{{\text{q}}^{\text{2}}}\text{+2q}\downarrow} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ -6q+3} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\text{ -6q+3}} \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \\
\end{align}\]
We can see that we have obtained the remainder as\[0\], so we can conclude that \[2q-1\] is a factor of \[4{{q}^{3}}-6{{q}^{2}}-4q+3\].

Note: We can check that if we have obtained the answer correctly or not by using the Euclid division algorithm i.e. \[\text{a=bq+r}\].
From the question, we have
\[\begin{align}
  & \text{a=}4{{q}^{3}}-6{{q}^{2}}-4q+3 \\
 & b=2q-1 \\
 & q=\text{2}{{\text{q}}^{\text{2}}}\text{- 2q-3} \\
 & r=0 \\
\end{align}\]
Upon substituting these values, let us check if we obtain the same polynomials on both the sides.
\[\begin{align}
  & \Rightarrow \text{a=bq+r} \\
 & \Rightarrow \left( 4{{q}^{3}}-6{{q}^{2}}-4q+3 \right)=\left( 2q-1 \right)\left( \text{2}{{\text{q}}^{\text{2}}}\text{- 2q-3} \right)+0 \\
 & \Rightarrow \left( 4{{q}^{3}}-6{{q}^{2}}-4q+3 \right)=4{{q}^{3}}-4{{q}^{2}}-6q-2{{q}^{2}}+2q+3 \\
 & \Rightarrow \left( 4{{q}^{3}}-6{{q}^{2}}-4q+3 \right)=4{{q}^{3}}-6{{q}^{2}}-4q+3 \\
\end{align}\]
We can see that we have obtained the same polynomials on both sides of the equation.
Hence proved.