Find the product of -3xyz 4 by 9 x -27 by 2 x z and verify
Answers
Answer:
earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are:
x + 3, 2y – 5, 3x2, 4xy + 7 etc.
You can form many more expressions. As you know expressions are formed from variables and constants. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7.
We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0,5/2 ,-7/3 etc.; actually countless different values.
The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc. Find the value of the expression 2y – 5 for the other given values of y.
Number line and an expression:
Consider the expression x + 5. Let us say the variable x has a position X on the number line;
X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P, 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on. What about the position of 4x and 4x + 5?
The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C.
TRY THESE
We have discussed in the above paragraphs a number of activities which we normally consider to be work in day-to-day life. For each of these activities, ask the following questions and answer them:
Give five examples of expressions containing one variable and five examples of expressions containing two variables.
Show on the number line x, x – 4, 2x + 1, 3x – 2.
2 Terms, Factors and Coefficients
Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5.
The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5.
TRY THESE
Identify the coefficient of each term in the expression x2y2 – 10x2y + 5xy2 – 20.
3 Monomials, Binomials and Polynomials
Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one.
Examples of monomials: 4x2, 3xy, –7z, 5xy2, 10y, –9, 82mnp, etc.
Examples of binomials: a + b, 4l + 5m, a + 4, 5 –3xy, z2 – 4y2, etc.
Examples of trinomials: a + b + c, 2x + 3y – 5, x2y – xy2 + y2, etc.
Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc.
TRY THESE
Classify the following polynomials as monomials, binomials, trinomials. – z + 5, x + y + z, y + z + 100, ab – ac, 17
Construct
3 binomials with only x as a variable;
3 binomials with x and y as variables;
3 monomials with x and y as variables;
2 polynomials with 4 or more terms.
4 Like and Unlike Terms
Look at the following expressions:
7x, 14x, –13x, 5x2, 7y, 7xy, –9y2, –9x2, –5yx
Like terms from these are:
(i) 7x, 14x, –13x are like terms. (ii) 5x2 and –9x2 are like terms.
(iii) 7xy and –5yx are like terms.
Why are 7x and 7y not like?
Why are 7x and 7xy not like?
Why are 7x and 5x2 not like?
TRY THESE
Write two terms which are like
7xy
4mn2
2l
5 Addition and Substraction of Algebraic Expressions
In the earlier classes, we have also learnt how to add and subtract algebraic expressions.
For example, to add 7x2 – 4x + 5 and 9x – 10, we do
7x2 – 4x + 5
+ 9x – 10
7x2 + 5x – 5
Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown.
Thus 5 + (–10) = 5 –10 = –5. Similarly, – 4x + 9x = (– 4 + 9)x = 5x. Let us take some more examples.
Example 1
Add: 7xy + 5yz – 3zx, 4yz + 9zx – 4y , –3xz + 5x – 2xy.
Solution
Writing the three expressions in separate rows, with like terms one below the other, we have
7xy + 5yz –3zx
+ 4yz + 9zx – 4y
+ –2xy– 3zx + 5x (Note xz is same as zx)
5xy + 9yz + 3zx + 5x – 4y
Thus, the sum of the expressions is 5xy + 9yz + 3zx + 5x – 4y. Note how the terms, – 4y in the second expression and 5x in the third expression, are carried over as they are, since they have no like terms in the other expressions.
Example 2