Question

State if True of False
Check whether the polynomial \[p\left( y \right) = 2{y^3} + {y^2} + 4y - 15\] is a multiple of \[\left( {2y - 3} \right)\]

Answer 1

Hint: Here we will use the long division method to check whether the given polynomial is a multiple of the given monomial or not.
If the remainder comes out to be zero then it is a multiple otherwise it is not a multiple.

Complete step-by-step answer:
Step 1:- First of all we will write the polynomial in descending order. If any of the terms are missing, then we will use a zero to fill in the missing term .
Step 2:- Then we will divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol.
Step 3:-Then we will multiply the answer obtained in the previous step by the polynomial in front of the division symbol.
Step 4:- then we will subtract and bring down the next term.
Step 5:- Repeat the steps 2,3 and 4 until there are no more terms to bring down.
Step 6:- Hence we will get the final answer. The term remaining after the last subtract step is the remainder.

Step by step solution:-
The given polynomial is:-
\[p\left( y \right) = 2{y^3} + {y^2} + 4y - 15\]
And the divisor monomial is:-
\[\left( {2y - 3} \right)\]
Now let us apply the long division method to check whether the polynomial is the multiple of the monomial or not.
Step 1:- First of all we will write the polynomial in descending order. If any of the terms are missing, then we will use a zero to fill in the missing term. In this case, the problem is already in the correct form.
         \[{\text{ }}\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15\]
 Step 2:- Now we will divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. In this case, we have \[2{y^3}\] divided by \[2y\] which is \[{y^2}\].
            \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2}\]
Step 3:- Now we will multiply the answer so obtained in the previous step by the polynomial in front of the division symbol. In this case we need to multiply by \[{y^2}\] and \[\left( {2y - 3} \right)\]
            \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2}\]
           \[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
Step 4:- Now subtract and bring down the next term.
           \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2}\]
           \[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                         \[{\text{4}}{{\text{y}}^2} + 4y\]
Step 5:- Again we will divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. In this case, we have \[{\text{4}}{y^2}\] divided by \[2y\]which is\[2y\].
              \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2} + 2y\]
          \[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                          \[{\text{4}}{{\text{y}}^2} + 4y\]
Step 6:- Now we will multiply the answer obtained in the previous step by the polynomial in front of the division symbol. In this case, we need to multiply \[2y\] and \[\left( {2y - 3} \right)\]
             \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2} + 2y\]
          \[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                          \[{\text{4}}{{\text{y}}^2} + 4y\]
                      \[{\text{ }} - \left( {4{y^2} - 6y} \right)\]
Step 7:- Again subtract and bring down the next term.
             \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2} + 2y\]
          \[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                          \[{\text{4}}{{\text{y}}^2} + 4y\]
                      \[{\text{ }}\underline { - \left( {4{y^2} - 6y} \right){\text{ }}} {\text{ }}\]
                                  \[10y - 15\]

Step 8:- Now again divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. In this case, we have \[10y\] divided by \[2y\] which is 5.
             \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2} + 2y + 5\]
\[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                          \[{\text{4}}{{\text{y}}^2} + 4y\]
                      \[{\text{ }}\underline { - \left( {4{y^2} - 6y} \right){\text{ }}} {\text{ }}\]
                                  \[10y - 15\]

Step 9:- Now we will multiply the answer obtained in the previous step by the polynomial in front of the division symbol. In this case, we need to multiply 5 and\[\left( {2y - 3} \right)\].
             \[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2} + 2y + 5\]
\[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                          \[{\text{4}}{{\text{y}}^2} + 4y\]
                      \[{\text{ }}\underline { - \left( {4{y^2} - 6y} \right){\text{ }}} {\text{ }}\]
                                  \[10y - 15\]
                                  \[10y - 15\]
Step 10:- Then subtract and finally we have no terms to bring down.
\[\underline {{\text{ }}} \]
\[2y - 3)2{y^3} + {y^2} + 4y - 15{\text{ }}({y^2} + 2y + 5\]
\[\underline {{\text{ }} - {\text{ }}\left( {2{y^3}{\text{ - }}3{y^2}} \right){\text{ }}} \]
                          \[{\text{4}}{{\text{y}}^2} + 4y\]
                      \[{\text{ }}\underline { - \left( {4{y^2} - 6y} \right){\text{ }}} {\text{ }}\]
                                  \[10y - 15\]
                                  \[\underline {10y - 15} \]
                                      \[{\text{0}}\]
Now since we got the remainder as zero
Therefore the polynomial \[p\left( y \right) = 2{y^3} + {y^2} + 4y - 15\] is a multiple of \[\left( {2y - 3} \right)\]

Hence the given statement is true.

Note: Students should note that a polynomial which is to be divided should have a higher degree of the variable than the polynomial which is the divisor.
Also, when the remainder on dividing a polynomial by another polynomial comes out to be zero then it means that it is a multiple of that polynomial.