Ch 5 Algebraic Expressions

Ch 5 Algebraic Expressions

Learning outcomes

After studying this chapter, you will be able to

# generate algebraic expression involving one or two variables /unknowns and operations .

# Identify constants, coefficient, powers ,like and unlike terms and degree of an expression (exponents ≤ 3) # add and subtract algebraic expressions (coefficients should be integers)

Terms related to algebraic expressions

Variable – A variable is a letter or symbol used to represent a value that can change. Eg a, b,c ,etc Constant – A constant is a value that does not change eg 1,2,897,-2343 ,etc

Algebraic expression- An algebraic expression is a combination of variables and constants connected by mathematical operators I.e. +,-,×,÷. Eg. 5 y², 20xy – 17x +5

Term – A term is either a single number or variable or numbers and variables multiplied together. Terms are separated by +

or – sign. Eg 5y + x, 20xy – 17x +5

Coefficients- Any of the factors of a term is called the coefficients of the product of the remaining factors. Eg 3 x coefficient of 3 is x or coefficient of x is 3.

Like and unlike terms:- The terms whose variable parts are exactly same meaning , meaning they have the same algebraic factors, are called like terms and the terms whose variable parts, i.e whose algebraic factors are different are called unlike terms.

Eg 5p and -9p are like terms 5x and 7y are unlike terms Types of algebraic expressions

Monomials:- Expression having one single term with whole numbers exponents are called monomials.eg -3x, 5y, etc

Binomials:- A binomial is an algebraic expression containing a sum or difference of exactly two monomials or terms. Eg x-2y, 2a+9b , etc

Trinomials:- A trinomials is an algebraic expressions containing the sum or difference of exactly three monomials or terms . eg x-2y+3z, a+5b+7c, etc

Polynomials :- A polynomial in two or more variables is an algebraic expressions containing two or more variables with whole number exponents.

Degree of polynomials:- A degree of polynomial is the highest power of the variable present in the polynomial. eg 2x2 _{– xy +9y}2 _{The degree of polynomial is 2}

Self Practice 5A

g. 17z h. x + _3𝑥1 -2

Polynomial Non polynomial

a. 5-7x _{c. 2y}2_{- xy +}5 𝑦 -3 b. y3_+16y2_-11 d. 2- _𝑥9 e. 8 -3x +4x2 _-6x3 h. x + _3𝑥1 -2 f. 19 g. 17 z

2. State the numerical coefficient and degree of each monomial

Equation Numerical coefficient Degree of polynomial

a. 45 Non polynomial 0 b. 4x2 y Non polynomial c. 42x-1/4 _{Non polynomial} d. 9b 9 3 e. 8pq 8 2 f. -12m2_{n -12 3} g. 5 T2 Non polynomial h. Xy2 _{1 3} i. -7d3 _{-7 3} j. -25x2_{y -25 3}

3. Select the like terms in each polynomial, if any a. 7x-3x+8; like terms are 7x and -3x

b. a+ b + 2a – c; like terms are a and 2a c. a + 3b – c + 5b; like terms are 3b and 5b

d. 2xy-5xz+4yz-yz; like terms 4yz and -yz

e. mn²+ m²n² – 2mn²+ 6 m²n²;like terms mn² and – 2mn²; m²n² and 6 m²n² f. 2ab²-7a²b+9a²b²+ 8 a²b; like terms -7a²b and 8a²b

g. 8(r+s)-3(r+s)+5 (r-s); like terms 8 (r+s) and -3(r+s) 4. State the degree of each polynomial.

d. 8x – 5y²+11x³; degree 3

e. 0.5 x²y + 0.25 xy²+6 x²+0.75; degree 3

5. Determine whether the given polynomials are minimal, binomial , trinomials or none of these. Then state the degree and coefficients of the polynomials.

a. 5c²d= monomial, degree 3 ,coefficient 5

b. 12x²+ d = binomial, degree 2 , coefficient 12 and 1 c. 15 m²-2p+5= trinomial ,degree 3 , coefficient 15 and -2 d. 8x -5 y²+11x³= trinomial, degree 3; coefficient 8 ,-5 and 11

f. -7 a³+ 5 a²b -ab²+ 10b³= none of these; degree 3; coefficient-7, 5,-1 and 10 6 . Arrange the given polynomials in order of decreasing degree in x.

a. 4x+x²-8= x²+4x-8

b. 8x²-3x -12 +6x³= 6x³+8x²-3x -12 c. x²y-3 y³+4x³-2xy²= 4x³+x²y-2xy²-3y³

d. 2x²y -8xy³+x³-y³+3xy= x³+2x²y-8xy³ +3xy -y³ 7. Write each polynomial in standard form. a. 17 – 3x² + x + 8 x³= 8x³-3x²+x+17 b. 20 y³ -13y -3 y² +1= 20y³ -3y²-13y +1 8. Evaluate if a=2,b=1 and c= 10 a. C³= 10×10×10=1000 b. (2b)²= = (2×1)²= 2×2=4 c. (c+b)²= (10+1)²= 11×11=121 d. (a-3)²= (2-3)²= (-1)×(-1)=1 e. (-3c)³= (-3×10)×(-3×10)×(-3×10)= -27000 f. 3a³/b= 3×2³/1=24 g. 9(a²-b²)= 9 (2²-1²)= 9 (4-1)= 9×3=27 h. 3b (a³_c)= 3×1(2³-10)= 3×(-2)= -6 I. a³+b²+c² = 2³+1²+10²= 8+1+100=109 j. (-a-b+c)= ( -2-1+10)²=(-7)²= 49

9. Write the algebraic expression for the following situations. a. Rita’s age is 4 less than 4 times her daughter’s age = 4x-4

b. The expression for a 3 digit number, when the hundreds digit is a, the tens digit is b and the units digit is c. = 100a+10b+c

c. The perimeter if an isosceles triangle whose equal sides are of length p cm each and unequal side is q cm .= 2p+q

d. The speed of a bus is zm/s . Write its speed in km/hr = 1km = 1000m and 1hr = 3600 sec; z3600÷1000 = 18/5zkm/hr

e. A pair of shoes cost rs x . The cost of dress is rs. 200 more than 5 times the cost of shoes.= 5x + 200

Addition of algebraic expressions

Additions of monomials:-

Eg: (i) 7a +3a =10a (ii) p+23p = 24p

Addition of polynomials

Polynomials can be added in either horizontal or vertical form Eg. 7x+19 and 2x-5 Horizontal form 7x+19+2x-5 = 7x+2x +19-5 9x +14 Vertical form

Step 1 : Write the polynomial one below the other aligning the like terms Step 2 : Add columnwise

Here, 7x+2x=9x and 19+(-5) = 14 7x+19 Thus the sum is 9x +14 + 2x -5

9x+14 Self Practice 5 B

= 8m2_n-5m2_n =3m2_n 2. Simplify a. –xy2_-xy2_{- xy}2 =. –xy2_-xy2_{- xy}2 = - 3 xy2

b. abc – abc + abc = 0+ abc

= abc

c. 12xy + 3xy – 4xy = 15xy – 4xy = 11xy d. ab – 7ab +10ab = -6ab + 10ab = 4ab e. -3 – 2m – 4 +7m = -3-4 -2m +7m = -7 +5m

+ 3x + 10 11x +15

c. 2xy – yz , 5xy + 7yz 2xy – yz + 5xy+ 7yz 7 xy +6yz d. 7m2 _{+ 9 n}2 _{, - 4m}2 _{– 2n}2 7m2 _{+ 9 n}2 - 4m2 _{– 2n}2 3m2 _{+ 7n}2 e. 11x – 6y, 4x + 5y, 7x – 2y 11x – 6y 4 x + 5y + 7x - 2y 22x – 3y f.2x2 _{+ 6x – 9, - 5x}2 _{+ 7x – 1, 6x}2 _{+ 13x + 21} 2x2 _{+ 6x – 9} - 5x2 _{+ 7x – 1} + 6x2 _{+ 13x + 21} 3x2 _+26x+11

g. 5a2 _+6ab+7b2 _{, 6a}2_{- 2ab+ 3b}2 _{, -4a}2 _{–ab -2b}2

5a2 _+6ab+7b2

6a2_{- 2ab+ 3b}2

-4a2 _{–ab - 2b}2

4. Add the following without writing in column (horizontally) a. 2a-4b+3c, -a+7b-c = 2a -4b+3c-a+7b-c =2a-a-4b+7b+3c-c = a+3b+2c b. 4m-5n+3p, -3m+3n-2p = 4m-5n+3p-3m+3n-2p = 4m-3m-5n+3n+3p-2p =m-2n+p c. 3ac+5b², -ac+b² = 3ac+5b2_-ac+b2 = 3ac-ac+5b2_+b2 = 2ac+6b2 d. 4a-b-c+1,2a+4b+4c-8 = 4a-b-c+1+2a+4b+4c-8 =4a+2a-b+4b-c+4c+1-8 =6a+3b+3c-7 e. 5x²+8x-4, 3x²-4x+11 = 5x²+8x-4+ 3x²-4x+11 =5x2_+3x2_{+ 8x-4x+3x}2 _-4+11 = 8x2 _+4x+7 f. 7m+8n-3mn,9n+7mn-3m = 7m+8n-3mn+9n+7mn-3m = 7m-3m+8n+9n-3mn+7mn =4m+17n+4mn Q5. Simplify a. 14x²-2xy+y²+5xy-2y²+5+x²+3y²+7 = 14x2_+x2 _{-2xy+5xy +y}2_-2y2_+3y2₊₅₊₇ = 15x2 _+3xy+2y2₊₁₂

b. 7-y³+y+9y²+2y³+6+y²+5y-19y+3y³-2y²+15 =7+6+15-y3_+2y3_+3y3_+9y2_-2y2_+y2_+y+5y-19y = 28+4y3_+8y2_-13y

Subtraction of algebraic expressions

Method

1. To subtract like terms , change the sign of the subtrahend mentally and then workout as in addition Eg 5a from 9a

9a-5a=4a

2. In case of unlike terms , the difference can only be indicated by reversing the sign of the subtrahend Eg. 5y from 3x

= 3x-5y

Subtract 2x-y from 4x-2y+7

-2a +5a=3a c. 3b2 _{from -5b}2 -5b2_-3b2 =-8b2 d. -2xy from 0 0-(-2xy)= 2xy e. -8cd from 8cd =8cd-(-8cd) = 8cd+8cd =16cd

f. 3(x+y) from 4(x+y) =4(x+y)-3(x+y) =(x+y)

d. 9x2_{y -11} - 2 x2_{y -5} 11x2 _-6 e. 2m3 _{– 9n}2 4m3 _{– 11n}2 - 2m3 _{+ 2n}2 f. 4x2_{y – 5xy}2 4x2_{y – 5xy}2 0

3. Arrange the following in columns and subtract a. 7p +2q +5c from 9p +4q +7c

9p +4q+ 7c 7p + 2q +5c 2p + 2q +2c

b. 3x2 _{– 8xy – 9y}2 _{from 3x}2 _{– 8xy – 9y}2

d. -2p3 _{+ 7p}2_{- 20+3p from 0}

0

-2p3 _{+ 7p}2_{- 20+3p}

2p3 _{- 7p}2_{+ 20-3p}

e. 4ax2 _{+7bx -9t +6 from 8ax}2 _{-2bx +10t -4}

8ax2 _{-2bx +10t -4}

4ax2 _{+7bx -9t + 6}

4ax2 _{-9bx +19t -10}

4. Subtract the following without writing in vertical form a. 2x +3y from 7x+9y

= 16- 8p + 2p

_-4p

_+7+p

= 23+2p

_-4p

_-7p

e. 4xy- 3x

_{y +7xy}

_{-3 from 27- 7xy}

_+11xy

= 27- 7xy

_{+11xy – (4xy- 3x}

_{y +7xy}

_-3)

= 27- 7xy

_{+11xy – 4xy+ 3x}

_{y -7xy}

₊₃

= 30 -14xy

_+3x

_{y +7xy}

f. The sum of x

_{-2xy +y}

_{and x}

_{+3xy + y}

_{from x}

_{-6xy +4y}

sum x

_{-2xy +y}

_{+ x}

_{+3xy + y}

=2x

_+xy

_+2y

Subtract x

_{-6xy +4y}

_–(2x

_+xy

_+2y

₎

= x

_{-6xy +4y}

_–2x

_-xy

_-2y

_{= - x}

_{-7xy + 2y}

b. 5x

_{-8 +7x}

_{y -7xy +10 xy}

_+11-2x

_-5x

_{y +9xy -6xy}

= 5x

_{- 2x}

_+7x

_{y- 5x}

_{y +10xy}

_{- 6xy}

_{-7xy+9xy -8 +11}

= 3x

_+2x

_{y +4xy}

_+2xy+3

6. The area of a square is 4x

_{-2xy +6y}

_{square units. A triangle inside has an area of}

-5xy +4y

_-3x

_{square units .Find the area of shaded portion.}

Sol. Area of square

=4x

_{-2xy +6y}

Area of triangle

=-3x

_-5xy+4y

Area of shaded part

= 7x

_{+3xy +2y}