5.1 Algebraic Expressions - Mathematics

5.1 Algebraic Expressions

5.1.1 Algebraic terms in Two or More Unknowns

1. An algebraic term in two or more unknowns is the product of the unknowns with a number.

Example 1:

4a³b = 4 × a × a × a× b

2. The coefficient of an unknown in the given algebraic term is the factor of the unknown.

Example 2:

7ab: Coefficient of ab is 7.

3. Like algebraic terms are algebraic terms with the same unknowns.

5.1.2 Multiplication and Division of Two or More Algebraic Terms

The coefficients and the unknowns of algebraic terms can be multiplied or divided altogether.

Example 3:

Calculate the product of each of the following pairs of algebraic terms.

(a) 5ac × 2bc

(b) –6xy × 5yz

(c) 20 x y \times (- \frac{2}{5} x^{2} y)

Solution:

(a)

5ac × 2bc = 5 × a × c × 2 × b × c = 10abc ²

(b)

–6xy × 5yz = –6 × x × y × 5 × y × z = –30 xy²z

(c)

\begin{array}{l} 20 x y \times (- \frac{2}{5} x^{2} y) \\ = 20^{4} x y \times (- \frac{2}{5}) \times x \times x \times y \\ = - 8 x^{3} y^{2} \end{array}

Example 4:

Find the quotients of each of the following pairs of algebraic terms.

\begin{array}{l} (a) \frac{42 x y z}{7 x y} \\ (b) \frac{12 x y^{2}}{18 x y} \end{array}

(c) 35 p^{2} q r^{2} \div 30 p r

Solution:

\begin{array}{l} (a) \frac{42 x y z}{7 x y} \\ = \frac{42^{6} x \times y \times z}{7 x \times y} = 6 z \end{array}

\begin{array}{l} (b) \frac{12 x y^{2}}{18 x y} \\ = \frac{12^{2} x \times y \times y}{18^{3} x \times y} = \frac{2}{3} y \end{array}

\begin{array}{l} (c) 35 p^{2} q r^{2} \div 30 p r \\ = \frac{35^{7} p \times p \times q \times r \times r}{30^{6} p \times r} \\ = \frac{7}{6} p q r \end{array}

5.1.3 Algebraic Expressions

An algebraic expression contains one or more algebraic terms. These terms are separated by a plus or minus sign.

Example 5:

7 – 6a² b + c is an algebraic expression with 3 terms.

5.1.4 Computation Involving Algebraic Expressions

Computation Involving Algebraic Expressions:

(a) 2(3a – 4) = 6a – 8

(b) (15a – 9b) ÷ 3 = 5a – 3b

(c) (6a – 2) – (9 + 4a)

= 6a – 4a – 2 – 9

= 2a – 11

(d) (a² b – 5ab²) – (6a² b – 4abc – 6ab²)

= a² b – 6a² b – 5ab²– (– 6ab²) – (– 4abc)

= –5a² b + ab² + 4abc