Simplify the following expression and find the product when x = -2 and y = -1.
$\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$
Answer
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Hint: The given problem is related to simplification of polynomials. Try to express the given terms as sum or difference of two or more terms by multiplying the given terms.
Complete step-by-step answer:
Before proceeding with the solution, first, we will understand the concept of the product of two polynomials. We will consider the two polynomials $(ax+by)$ and $(px+qy+r)$. To find the product of the two polynomials, we have to multiply each term of the second polynomial by each term of the first polynomial. So, the product of the two polynomials $(ax+by)$ and $(px+qy+r)$ is given as $(ax+by)(px+qy+r)=ax\left( px+qy+r \right)+by\left( px+qy+r \right)$.
$=ap{{x}^{2}}+aqxy+arx+bpxy+bq{{y}^{2}}+bry$
$=ap{{x}^{2}}+bq{{y}^{2}}+xy\left( aq+bp \right)+r\left( ax+by \right)$
Now, coming to the question, we are asked to simplify the product $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$.
So, $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$can be written as $2x{{y}^{2}}\left( 3x+y-4xy \right)-{{x}^{2}}{{y}^{2}}\left( 3x+y-4xy \right)$.
Now, we will multiply each term to get the product. On multiplying each term , we get $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-8{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}-{{x}^{2}}{{y}^{3}}+4{{x}^{3}}{{y}^{3}}$.
$=6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$
Hence, the product $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$ is simplified to $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$.
Now, we need to find the value of the product when $x=-2$ and $y=-1$. To find the value of the product, we will substitute $x=-2$ and $y=-1$ in $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$. On substituting $x=-2$ and $y=-1$ in $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$, we get:$6{{\left( -2 \right)}^{2}}{{\left( -1 \right)}^{2}}+2\left( -2 \right){{\left( -1 \right)}^{3}}-9{{\left( -2 \right)}^{2}}{{\left( -1 \right)}^{3}}-3{{\left( -2 \right)}^{3}}{{\left( -1 \right)}^{2}}+4{{\left( -2 \right)}^{3}}{{\left( -1 \right)}^{3}}$.
Now, we know ${{\left( -1 \right)}^{2}}=1$ , ${{\left( -1 \right)}^{3}}=-1$, ${{\left( -2 \right)}^{2}}=4$ and ${{\left( -2 \right)}^{3}}=-8$.
So, the value of the product will be equal to $\left( 6\times 4\times 1 \right)+\left( 2\times \left( -2 \right)\times \left( -1 \right) \right)-\left( 9\times 4\times \left( -1 \right) \right)-\left( 3\times \left( -8 \right)\times 1 \right)+\left( 4\times \left( -8 \right)\times \left( -1 \right) \right)$.
$=24+4+36+24+32$
$=120$
Hence, the value of $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$ when $x=-2$ and $y=-1$ is $120$.
Note: While substituting the values of $x$ and $y$ in the simplified expression, make sure that sign mistakes do not occur. Sign mistakes are very common and due to such mistakes, students can end up getting a wrong answer. So, such mistakes should be avoided in every possible case.
Complete step-by-step answer:
Before proceeding with the solution, first, we will understand the concept of the product of two polynomials. We will consider the two polynomials $(ax+by)$ and $(px+qy+r)$. To find the product of the two polynomials, we have to multiply each term of the second polynomial by each term of the first polynomial. So, the product of the two polynomials $(ax+by)$ and $(px+qy+r)$ is given as $(ax+by)(px+qy+r)=ax\left( px+qy+r \right)+by\left( px+qy+r \right)$.
$=ap{{x}^{2}}+aqxy+arx+bpxy+bq{{y}^{2}}+bry$
$=ap{{x}^{2}}+bq{{y}^{2}}+xy\left( aq+bp \right)+r\left( ax+by \right)$
Now, coming to the question, we are asked to simplify the product $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$.
So, $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$can be written as $2x{{y}^{2}}\left( 3x+y-4xy \right)-{{x}^{2}}{{y}^{2}}\left( 3x+y-4xy \right)$.
Now, we will multiply each term to get the product. On multiplying each term , we get $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-8{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}-{{x}^{2}}{{y}^{3}}+4{{x}^{3}}{{y}^{3}}$.
$=6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$
Hence, the product $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$ is simplified to $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$.
Now, we need to find the value of the product when $x=-2$ and $y=-1$. To find the value of the product, we will substitute $x=-2$ and $y=-1$ in $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$. On substituting $x=-2$ and $y=-1$ in $6{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}-9{{x}^{2}}{{y}^{3}}-3{{x}^{3}}{{y}^{2}}+4{{x}^{3}}{{y}^{3}}$, we get:$6{{\left( -2 \right)}^{2}}{{\left( -1 \right)}^{2}}+2\left( -2 \right){{\left( -1 \right)}^{3}}-9{{\left( -2 \right)}^{2}}{{\left( -1 \right)}^{3}}-3{{\left( -2 \right)}^{3}}{{\left( -1 \right)}^{2}}+4{{\left( -2 \right)}^{3}}{{\left( -1 \right)}^{3}}$.
Now, we know ${{\left( -1 \right)}^{2}}=1$ , ${{\left( -1 \right)}^{3}}=-1$, ${{\left( -2 \right)}^{2}}=4$ and ${{\left( -2 \right)}^{3}}=-8$.
So, the value of the product will be equal to $\left( 6\times 4\times 1 \right)+\left( 2\times \left( -2 \right)\times \left( -1 \right) \right)-\left( 9\times 4\times \left( -1 \right) \right)-\left( 3\times \left( -8 \right)\times 1 \right)+\left( 4\times \left( -8 \right)\times \left( -1 \right) \right)$.
$=24+4+36+24+32$
$=120$
Hence, the value of $\left( 2x{{y}^{2}}-{{x}^{2}}{{y}^{2}} \right)\left( 3x+y-4xy \right)$ when $x=-2$ and $y=-1$ is $120$.
Note: While substituting the values of $x$ and $y$ in the simplified expression, make sure that sign mistakes do not occur. Sign mistakes are very common and due to such mistakes, students can end up getting a wrong answer. So, such mistakes should be avoided in every possible case.
Last updated date: 23rd Sep 2023
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