Algebraic Expressions - Exercise 13(A)

Identify monomials, binomials and trinomials from the following algebraic expressions.

(i) 7p² × a³b

(ii) 5 + 3x³y³z²

(iii) 3x² ÷ p

(iv) $\dfrac{3a + 2b - 5c}{7}$

(v) xy + yz - zx

(vi) ax² + bx × y²

Answer

(i) 7p² × a³b

Multiplication combines these into a single unit. There are no + or - signs separating them.

Number of terms: 1

Hence, it is a monomial.

(ii) 5 + 3x³y³z²

The + sign separates the constant '5' from the variable term.

Number of terms: 2

Hence, it is a binomial.

(iii) 3x² ÷ p

Division results in a single algebraic term : $\dfrac{3\text x^2}{\text p}$

So,

Number of terms: 1

Hence, it is a monomial.

(iv) $\dfrac{3a + 2b - 5c}{7}$

While the entire expression is over a single denominator, the numerator contains three distinct terms (3a, 2b, and -5c) separated by + and -.

⟹ Number of terms: 3

Hence, it is a trinomial.

(v) xy + yz - zx

There are three distinct products separated by + and -.

So,

Number of terms: 3

Hence, it is a trinomial.

(vi) ax² + bx × y²

The multiplication combines bx and y² into one term (bxy²). The only separator is the + sign.

Term 1: ax²

Term 2: bxy²

Number of terms: 2

Hence, it is a binomial.

Write down the numerical as well as literal coefficient of each of the following monomials:

(i) -2p³q²

(ii) $\dfrac{5}{9}xyz^2$

(iii) $-\dfrac{3ab^2}{2}$

(iv) $\dfrac{3}{a^2}$

(v) $\dfrac{2ab}{c}$

(vi) $-\dfrac{2x^2}{yz}$

Answer

(i) -2p³q²

Numerical coefficient = −2, Literal coefficient = p³q²

(ii) $\dfrac{5}{9}xyz^2$

Numerical coefficient = $\dfrac{5}{9}$, Literal coefficient = xyz²

(iii) $-\dfrac{3ab^2}{2}$

Numerical coefficient = $-\dfrac{3}{2}$, Literal coefficient = ab²

(iv) $\dfrac{3}{a^2}$

Numerical coefficient = 3, Literal coefficient = $\dfrac{1}{a^2}$

(v) $\dfrac{2ab}{c}$

Numerical coefficient = 2, Literal coefficient = $\dfrac{ab}{c}$

(vi) $-\dfrac{2x^2}{yz}$

Numerical coefficient = -2, Literal coefficient = $\dfrac{x^2}{yz}$

In -5p²q³r⁴, write down the coefficient of:

(i) p²

(ii) 5pq²

(iii) p²q²

(iv) pqr

(v) -pq²r³

(vi) 5q³r

Answer

(i) p²

Coefficient of p²:

Divide the original expression by p² then

-5p²q³r⁴ ÷ p²

= -5q³r⁴

∴ Coefficient of p² is -5q³r⁴

(ii) 5pq²

Coefficient of 5pq²:

Divide the original expression by 5pq² then

-5p²q³r⁴ ÷ 5pq²

= -pqr⁴

∴ Coefficient of 5pq² is -pqr⁴

(iii) p²q²

Coefficient of p²q²:

Divide the original expression by p²q² then

-5p²q³r⁴ ÷ p²q²

= -5qr⁴

∴ Coefficient of p²q² is -5qr⁴

(iv) pqr

Coefficient of pqr:

Divide the original expression by pqr then

-5p²q³r⁴ ÷ pqr

= -5pq²r³

∴ Coefficient of pqr is -5pq²r³

(v) -pq²r³

Coefficient of -pq²r³:

Divide the original expression by -pq²r³ then

-5p²q³r⁴ ÷ -pq²r³

= 5pqr

∴ Coefficient of -pq²r³ is 5pqr

(vi) 5q³r

Coefficient of 5q³r:

Divide the original expression by 5q³r then

-5p²q³r⁴ ÷ 5q³r

= -p²r³

∴ Coefficient of 5q³r is -p²r³

Identify like terms in the following:

(i) a², b², -2a², c², 4a

(ii) 3x, 4xy, -yz, $\dfrac{1}{2}$zy

(iii) -2xy², x²y, 5y²x, x²z

(iv) abc, ab²c, acb², c²ab, b²ac, a²bc, cab²

Answer

(i) a², b², -2a², c², 4a

Since a² and -2a² have the same literal part

∴ Like terms are a² and -2a².

(ii) 3x, 4xy, -yz, $\dfrac{1}{2}$zy

Since -yz and $\dfrac{1}{2}$zy have the same literal part

∴ Like terms are -yz and $\dfrac{1}{2}$zy.

(iii) -2xy², x²y, 5y²x, x²z

Since -2xy² and 5y²x have the same literal part

∴ Like terms are -2xy² and 5y²x.

(iv) abc, ab²c, acb², c²ab, b²ac, a²bc, cab²

Since ab²c, acb², b²ac and cab² have the same literal part

∴ Like terms are ab²c, acb², b²ac and cab².

Generate algebraic expressions for the following:

(i) The product of a and b added to their sum.

(ii) The quotient of x by 8 is multiplied by y.

(iii) Thrice x added to y squared.

(iv) One-third of x multiplied by the sum of p and q.

(v) The number to be added to x + y to make it equal to p.

(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.

(vii) Five times m subtracted from the sum of p and thrice x.

(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.

(ix) The sum of 7 numbers each equal to p.

(x) The product of three numbers a, b and c subtracted from the sum of x and y.

Answer

(i) The product of a and b added to their sum.

Product of a and b = ab

Sum of a and b = (a + b)

∴ The expression is (a + b) + ab

(ii) The quotient of x by 8 is multiplied by y.

Quotient of x by 8 = $\dfrac{x}{8}$

Multiplied by y = $\dfrac{x}{8} \times y$

∴ The expression is $\dfrac{x}{8} \times y$

(iii) Thrice x added to y squared.

Thrice x = 3x

y squared = y²

∴ The expression is y² + 3x

(iv) One-third of x multiplied by the sum of p and q.

One-third of x = $\dfrac{1}{3}x$

Sum of p and q = (p + q)

∴ The expression is $\dfrac{1}{3}x(p + q)$

(v) The number to be added to x + y to make it equal to p.

Let number = ?

x + y + ? = p

⇒ ? = p − (x + y)

∴ The expression is p - (x + y)

(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.

Total length = p + q

Number of pieces = n

Length of each piece = $\dfrac{\text{Total length}}{\text{Number of pieces}}$

Length of each piece = $\dfrac{p + q}{n}$

∴ The expression is $\dfrac{p + q}{n}$

(vii) Five times m subtracted from the sum of p and thrice x.

Five times m = 5m

Sum of p and thrice x = (p + 3x)

∴ The expression is (p + 3x) - 5m

(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.

Difference of x and y = (x − y)

m times difference = m(x - y)

Sum of x and y = x + y

n times sum = n(x + y)

∴ The expression is n(x + y) - m(x - y)

(ix) The sum of 7 numbers each equal to p.

Adding p seven times is the same as 7 multiplied by p.

∴ The expression is 7p

(x) The product of three numbers a, b and c subtracted from the sum of x and y.

Product of a, b, c = abc

Sum of x and y = x + y

∴ The expression is (x + y) - abc

Identify which of the following expressions are polynomials. If so, write their degrees.

(i) $\dfrac{2}{3}x^4 - 7x + 15$

(ii) 8x² - 3x + 6√x + 1

(iii) 5x² - $\dfrac{2}{x}$ + 7

(iv) 9x²y² - 3xy² + 5x²y - 6x

(v) 6p⁴ - p³q² + pq³ + q⁴

(vi) 4x⁵ - 7x⁵y + 3xy⁴ + 8y⁵

(vii) ab² - $\dfrac{7}{a^2}$ + 5b² + 6

Answer

(i) $\dfrac{2}{3}x^4 - 7x + 15$

All powers of x are whole numbers.

The highest power of x is 4.

Yes, it is a polynomial. Degree = 4

(ii) 8x² - 3x + 6√x + 1

The term 6√x can be written as 6x^1/2, 1/2 is not a whole number.

∴ It is not a polynomial.

(iii) 5x² - $\dfrac{2}{x}$ + 7

The term $\dfrac{2}{x}$ can be written as 2x^-1, power of x is a negative integer.

∴ It is not a polynomial.

(iv) 9x²y² - 3xy² + 5x²y - 6x

All powers of x and y are whole numbers.

Degrees of terms:

9x²y² → 2 + 2 = 4

3xy² → 1 + 2 = 3

5x²y → 2 + 1 = 3

6x → 1

The highest degree is 4.

Yes, it is a polynomial. Degree = 4

(v) 6p⁴ - p³q² + pq³ + q⁴

All powers of p and q are whole numbers.

Degrees of terms:

6p⁴ → 4

p³q² → 3 + 2 = 5

pq³ → 1 + 3 = 4

q⁴ → 4

The highest degree is 5.

Yes, it is a polynomial. Degree = 5

(vi) 4x⁵ - 7x⁵y + 3xy⁴ + 8y⁵

All powers of x and y are whole numbers.

Degrees of terms:

4x⁵ → 5

7x⁵y → 5 + 1 = 6

3xy⁴ → 1 + 4 = 5

8y⁵ → 5

The highest degree is 6.

Yes, it is a polynomial. Degree = 6

(vii) ab² - $\dfrac{7}{a^2}$ + 5b² + 6

The term $\dfrac{7}{a^2}$ means 7a^-2, power of a is a negative integer.

∴ It is not a polynomial.