Multiplication of Algebraic Expressions
Let us solve some problems here based on the multiplication of different types of algebraic expressions.
1: Multiply 5x with 21y and 32z
Solution: 5x × 21y × 32z = 105xy × 32z = 3360xyz
We multiply the first two monomials and then the resulting monomial to the third monomial.
2: Find the volume of a cuboid whose length is 5ax, breadth is 3by and height is 10cz.
Solution:
Volume = length × breadth × height
Therefore, volume = 5ax × 3by × 10cz = 5 × 3 × 10 × (ax) × (by) × (cz) = 150axbycz
3: Multiply (2a2 + 9a + 10) by 4a.
Solution:
4a × (2a2 + 9a + 10)
= (4a × 2a2) + (4a × 9a) + (4a × 10)
= 8a3 + 36a2 + 40a
4: Simplify the below algebraic expression and obtain its value for x = 3.
x(x − 2) + 5
Solution: Given, x(x − 2) + 5, x = 3.
On simplifying the given expression, we get:
x2-2x+5
Now putting x = 3, we get;
= 32-2(3)+5
= 9 – 6 + 5
= 8
5: Simplify the below algebraic expression and obtain its value for y = −1
4y(2y − 6) – 3(y − 2) + 20
Solution: 4y(2y − 6) − 3(y − 2) + 20 for y = −1
Substituting the value of y = −1.
4 × −1((2 × −1) – 6) – 3(−1 − 2) + 20
= −4 (−2 − 6) − 3(−3) + 20
= 32 + 9 + 20 = 61.
Division of Algebraic Expressions
In the division of an algebraic expression, we cancel the common terms, which is similar to the division of numbers. Division of algebraic expressions involves the following steps.
- Step 1: Directly take out common terms or factories in the given expressions to check for the common terms.
- Step 2: Cancel the common term.
Note: Here, the common terms correspond to either of the following: constants, variables, terms, or just coefficients.
There are different types of division of algebraic expressions.
- Division of monomial by a monomial
- Division of polynomial by a monomial
- Division of polynomial by a polynomial
In any case, we first take out common terms from the given polynomials and then eliminate that common term/terms.
Division of Monomial by a Monomial
A monomial is a type of expression that has only one term. The correct method to perform the division of a monomial by another monomial is given below:
Consider an example, 27x3÷3x
Here 3x and 27x3 be the two monomials.
- Write their prime factorization. 27x3 ÷ 3x = 27×x×x×x/3×x
- Cancel the common term, which is 3x.
Thus, 27x3 ÷ 3x = 9x2
Division of Polynomial by a Monomial
A polynomial contains a few types of expressions, some of which are binomial, trinomial, or an equation with n-terms.
Now, let’s perform dividing polynomials by monomials.
(4y3 + 5y2 + 6y) ÷ 2y
Here, the trinomial is 4y3 + 5y2 + 6y, and the monomial is 2y.
- In trinomial, on taking the common factor 2y, it becomes: 4y3 + 5y2 + 6y = 2y(2y2 + (5/2)y + 3)
- Now, we do the division operation: {2y(2y2 + (5/2)y + 3)} ÷ 2y. Cancel 2y from the numerator and the denominator: (4y3 + 5y2 + 6y) ÷ 2y = 2y2 + (5/2)y + 3
Thus, (4y3 + 5y2 + 6y) ÷ 2y = 2y2 + (5/2)y + 3
Division of Polynomial by a Polynomial
Let us consider polynomials that divide polynomials for performing the division operation.
(7x2 + 14x) ÷ (x + 2)
Here, both polynomials exist in the binomial form.
- Take out the common factors. For the polynomial 7x2 + 14x, x is the common factor.
- So, consider “7x” as a common factor among them. Then it becomes, 7x2 + 14x = 7x(x+2)
- Now, (7x2 + 14x) ÷ (x + 2) = 7x(x + 2) / (x + 2)
- Eliminate (x+2) from the numerator and denominator, we get the solution for the long dividing polynomials as: (7x2 + 14x) ÷ (x + 2) = 7x
Thus, (7x2 + 14x) ÷ (x + 2) = 7x