Simplify 9t4 + 7t – 45t3 + 10t

Mathematics is a broad area of study with geometry, algebra, arithmetic, exponential, etc. Algebra is a mathematical branch of which is the study of symbols, letters, numbers, and their formulas. Algebraic expressions are the mathematical expressions that involve integer constants, variables, coefficient of variables, and power on the head of variables. These expressions are tied together with different algebraic operations.

Basic formulas of algebra

Algebra is about the study and manipulation of mathematical symbols. The expressions of algebra include known real numbers and unknown values. Hence, the operation is performed to determine these unknown quantities. Some of the basic standard formulas are derived to carry out the calculation. These formulas are listed below.

• a² – b² = (a – b)(a + b)
• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a² + b² = (a – b)² + 2ab.
• (a + b + c)² = a²+ b²+ c²+ 2ab + 2ac + 2bc.
• (a – b – c)² = a² + b² + c² – 2ab – 2ac + 2bc.
• a³ – b³ = (a – b) (a² + ab + b²)
• a³ + b³ = (a + b) (a² – ab + b²)

Types of algebraic operations on the basis of the number of variables

There are majorly three types of algebraic expressions on the basis of a number of variables. They are monomial expressions, binomial expressions, and polynomial expressions. Let’s take a look at their definitions in detail,

1. Monomial expressions: Monomial expressions are algebraic expression that only has one variable. For example: 5x, 2y,+1 are some monomial expressions.
2. Binomial expressions: Binomial expressions are algebraic expressions having two variables. For example: 2x + 3, x + y, etc are some binomial expressions.
3. Polynomial expressions: Polynomial expressions are algebraic expressions having multiple variables. For example: ab + bc + ca, x + xy + 1, etc are some polynomial expression.

Simplify: 9t⁴  +  7t – 45t³ + 10t

Solution:

Simplifying,

Given, algebraic expression

= 9t⁴ + 7t – 45t³ + 10t

• Step 1: Identify the like terms in the given equation. In the given equation, 7t and 10t are the like terms having the same variable ‘t’ and exponent power 1.
• Step 2: Arranging the like terms for simplification,

= 9t⁴ + 7t + 10t – 45t³

• Step 3: Combining the like terms. Here, we are adding the like terms 7t and 10t as per the mathematical operation attached to the terms.

= 9t⁴ + 17t – 45t³

Sample Problems

Question 1: Simplify 3a(4a – 5) + 3, if a = 3.

Solution:

Let’s simplify,

= 3a(4a – 5) + 3

= 12a² – 15a + 3

For a = 3

= 12(3)² – 15(3) + 3

= (12 × 9) – (15 × 3) + 3

= 108 – 45 + 3

= 66

Question 2: 12x² – 9x + 5x – 4x² – 7x + 10

Solution:

Let’s simplify,

= 12x² – 4x² + 5x – 9x – 7x + 10

= x²(12 – 4) + x(5 – 9 – 7) + 10

= x²(8) + x( -4 -7) + 10

= 8x² – 11x + 10

Question 3: Simplify (p² + 5)(q³ + 3)

Solution:

Let’s simplify,

= p²(q³ + 3) +5(q³ + 3)

= (p² × q³) + (p² × 3) + (5 × q³) + (5 × 3)

= p²q³ + 3p² + 3q³ + 15

Question 4: Simplify (a – 3)(a + 5)

Solution:

Let’s simplify,

= a(a + 5) -3(a + 5)

= a² + 5a – 3a – 15

= a² + 2a – 15