8类NCERT解决方案–第9章代数表达式和恒等式–练习9.1

问题1.确定下列各表达式的项及其系数。

Need to be Known:

• Expression: An Expression is the addition of terms. Example: 7y + z is an expression made up of two terms 7y and z.
• Term: Terms itself is a product of factors. Example: 7y + z, here terms are 7y (term 1) and z (term2) where term 1 is the product of two factors 7 and y . and term 2 is a single factor z.
• Coefficient: It is a numerical factor of the term. Example: 7y + z here term1 has a two factor 7 and y in which 7 is numerical factor hence it is a coefficient.

(i)5xyz ² – 3zy

解决方案：

1. Here terms are 5xyz² and -3zy
2. Here Coefficients are 5 and -3

(ii)1 + x + x ²

解决方案：

1. Here terms are 1, x, x²
2. Here Coefficients are 1, 1, 1

(iii)4x ² y ² – 4x ² y ² z ² + z ²

解决方案：

1. Here terms are 4x²y², –4x²y²z² and z²
2. Here Coefficients are 4, -4 and 1

(iv) 3 – pq + qr – rp

解决方案：

1. Here terms are -3, -pq, qr, -rp
2. Here Coefficients are 3, -1, 1 and -1

(v) x / 2 + y / 2-xy

解决方案：

1. Here terms are x/2, y/2 and -xy
2. Here Coefficients are 1/2, 1/2 and -1

(vi)0.3a – 0.6ab + 0.5b

解决方案：

1. Here terms are 0.3a, -0.6ab and 0.5b
2. Here Coefficients are 0.3, -0.6 and 0.5

问题2.将以下多项式分类为单项式，二项式和三项式。哪些多项式不适合这三个类别中的任何一个?

X + Y，1000，X + X ² + X ³ + X ^4，7 + Y + 5倍，2Y – 3Y ^2，2Y – 3Y ² + 4Y ^3，5× – 4Y + 3XY，4Z – 15Z ^2，AB + BC + CD + DA，PQR，p ² q + PQ ^2，2P + 2Q

Need to be Known:

• Monomial: Expression made up of one term is known as monomial. Example: pqr, z, ab etc. 
• Binomial: Expression made up of two terms is known as binomial. Example: 2y + 3z, 5x – 4y, etc.
• Trinomial: Expression made up of three terms is known as trinomial. Example: 2y + 3z + x, 5x – 4y + a, etc.
• Polynomial: Expression made up of one or more terms is known as a polynomial. Example: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc

解决方案：

In this question:

• Monomials: 1000, pqr
• Binomials: x + y, 2y – 3y², 4z – 15z², p²q + pq², 2p + 2q
• Trinomials: 7 + y + 5x, 2y – 3y²+ 4y³, 5x – 4y + 3xy
• and those which do not fit in these categories are:  x + x²+ x³+ x⁴, ab + bc + cd + da

问题3.添加以下内容

Need to be Known:

Like term: Terms having the same variable with their same power.

Note**: Terms those are like can only add and subtract each other. unlike terms cannot add and subtract each other. Example: 4x and 5x are like terms as they both have the same coefficient with the same power of x as 1.

Unlike term: Terms having different variable and may have different powers.

1. Example: 4x and 5y are unlike terms as they both have different variables
2. Example: 4x² and 5x³ are unlike terms as they both terms differ in the variable power.

(i)ab – bc，bc – ca，ca – ab

解决方案：

As here three expressions are given, we need to place expressions same terms below other expressions same term one by one then we will get like these: 

                                          ab – bc + 0

                                   +    0  + bc  –  ca

                                   +   -ab + 0  + ca

                                  _________________

                                         0  + 0  + 0                     <———-final expression

                                  _________________ 

this tends to 0.

(ii)a – b + ab，b – c + bc，c – a + ac

解决方案：

As here three expressions are given, we need to place expressions same terms below other expressions same term one by one then we will get like these: 

                         a – b + ab

                +      0 + b  + 0   – c + bc

                +      -a + 0  + 0  + c + 0 + ac

                       __________________________

                        0 + 0  + ab + 0 + bc + ac                   <—- final expression

                     _____________________________

this tends to ab + bc + ac.

(iii)2p ² q ² – 3pq + 4，5 + 7pq – 3p ² q ²

解决方案：

As here two expressions are given, we need to place expressions same terms below other expressions same term one by one then we will get like these: 

                               2p²q² – 3pq + 4

                   +  -3p²q² + 7pq +5

                        __________________

                        -1p²q² + 4pq + 9                <——– final expression

                      ____________________

this tends to -1p²q² + 4pq + 9.

(ⅳ)升² + M ^{2，M 2} + N ^{2，N 2} + L ^2，2LM + 200万+ 2NL

解决方案：

As here four expressions are given, we need to place expressions same terms below other expressions same term one by one then we will get like these: 

                        l² + m²

                   +  0 + m² + n²

                   +  l² + 0   + n²

                   +  0  + 0 + 0 + 2lm + 2mn + 2nl

                 _________________________________

                      2l² + 2m² + 2n² + 2lm + 2mn + 2nl             <———final expression

                   ___________________________________

as 2 is common among all terms then we can take it common this tends to  2( l² + m² + n² + lm + mn + nl)

问题4。

(a)从12a – 9ab + 5b – 3中减去4a – 7ab + 3b + 12

解决方案：

Here we need to substract the given expressions:

then,

                                 12a – 9ab + 5b – 3

                          (-)     4a – 7ab + 3b + 12

                             ________________________

in substraction sign of (-) expression gets altered as:

                            12a – 9ab + 5b – 3

                          – 4a + 7ab – 3b  – 12

                         ________________________

                              8a  – 2ab + 2b – 15                  <———final expression

                        __________________________

this tends to  8a  – 2ab + 2b – 15 .

(b)从5xy – 2yz – 2zx + 10xyz减去3xy + 5yz – 7zx

解决方案：

Here we need to subtract the given expressions:

then,

                        5xy – 2yz – 2zx + 10xyz

                   (-)  3xy + 5yz – 7zx  + 0

                    ___________________________

in substraction sign of (-) expression gets altered as:

                    5xy – 2yz – 2zx + 10xyz

                   -3xy – 5yz + 7zx  – 0

                ___________________________

                   2xy – 7yz + 5zx + 10xyz                       <———– final expression

               _____________________________

this tends to 2xy – 7yz + 5zx + 10xyz.

(c)从18 – 3p – 11q + 5pq – 2pq ² + 5p ² q^{减去4p 2} q – 3pq + 5pq ^{2 – 8p + 7q – 10}

解决方案：

Here we need to substract the given expressions:

then,           5p²q + 5pq – 2pq² – 3p – 11q + 18         

               (-) 4p²q – 3pq + 5pq² – 8p + 7q – 10

               __________________________________                

in substraction sign of (-) expression gets altered as:

                   5p²q  + 5pq – 2pq² – 3p – 11q + 18

                  -4p²q + 3pq – 5pq² + 8p – 7q  + 10

                 __________________________________        

                   p²q + 8pq  – 7pq² + 5p – 18q + 28              <———–final expression

                  _____________________________________             

this tends to  p²q + 8pq – 7pq² + 5p – 18q + 28