Question

Factorise (a) \(x^{2}+x\), (b) \(3 x^{2}+6 x\) (c) \(9 x^{2}-12 x\).

Short answer

Answer: The factorised forms are: (a) \(x(x+1)\), (b) \(3x(x+2)\), and (c) \(3x(3x-4)\).

Step-by-step solution

Identify the common factors of terms.

Look for the factors that are common to all the terms in the given expression.

Factor out the common factors.

Take the common factors outside the parenthesis and write the remaining expression inside. (a) \(x^{2}+x\)

Identify the common factors of terms.

The common factor for both terms is 'x'.

Factor out the common factors.

We take 'x' as a common factor from both terms and put the remaining expression in parenthesis: \(x(x+1)\) So, the factorised form of \(x^{2}+x\) is \(x(x+1)\). (b) \(3x^{2}+6x\)

Identify the common factors of terms.

The common factors for both terms are '3' and 'x'.

Factor out the common factors.

We take '3x' as a common factor from both terms and put the remaining expression in parenthesis: \(3x(x+2)\) So, the factorised form of \(3x^{2}+6x\) is \(3x(x+2)\). (c) \(9x^{2}-12x\)

Identify the common factors of terms.

The common factors for both terms are '3' and 'x'.

Factor out the common factors.

We take '3x' as a common factor from both terms and put the remaining expression in parenthesis: \(3x(3x-4)\) So, the factorised form of \(9x^{2}-12x\) is \(3x(3x-4)\).

Key concepts

Common Factors

A common factor is a number or variable shared by all terms in an expression. Finding common factors is a key step in factorization, especially in algebraic expressions. Let's break this down.

• For example, in the expression \(x^2 + x\), the terms are \(x^2\) and \(x\). They both share \(x\) as a factor, making it a common factor.
• Similarly, the expression \(3x^2 + 6x\) has the number 3 and the variable \(x\) as common factors.

To identify common factors:- Look for numbers and variables that can exactly divide all terms.- Consider both numerical coefficients and variables separately.By factoring out these common elements, we simplify expressions and often unlock further insights or simplification potential for solving equations.

Polynomials

Polynomials are algebraic expressions made up of terms created by variables and coefficients, using only the operations of addition, subtraction, multiplication, and whole-number exponents on the variables.

• Every term in a polynomial is a product of a coefficient and a power of a variable.
• For instance, in \(3x^2 + 6x\), '3' and '6' are coefficients while \(x^2\) and \(x\) are powers of the variable \(x\).

We classify polynomials by degree:- The degree of a term is determined by the exponent of the variable.- The polynomial's degree is the highest among its terms. Understanding polynomials is crucial since they form the basis for many algebraic concepts, including equations, functions, and graphing. Efficiently working with polynomials can involve simplifying, factoring, or expanding them as needed.

Elementary Algebra

Elementary algebra is the area of mathematics that introduces basic algebraic techniques and concepts. One of the foundational skills in elementary algebra is the ability to manipulate and simplify expressions, like factorization.

• Factorization involves rewriting expressions as the product of simpler factors.
• It's an essential skill that helps in solving algebraic equations and inequalities.

In exercises like the given one, elementary algebra focuses on locating common factors to simplify expressions. This focuses on simplifying by reversing the distributive property:- If \(ax + ay\) is an expression, it can be rewritten as \(a(x+y)\) by factoring out \(a\).- This transformation makes potential roots and solutions clearer, especially in equations. Grasping elementary algebraic techniques provides a strong foundation for future study in mathematics, preparing students for more complex topics such as quadratic equations, rational expressions, and beyond.