Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra → Polynomials ↓


Factorization of Polynomials


Factorization plays an important role in the study of polynomials in grade 10 maths. It is a technique used to break down a polynomial into a product of simpler polynomials. By multiplying these simpler polynomials with each other, we get the original polynomial back. Understanding how to factor polynomials can be extremely helpful in solving equations, simplifying expressions, and much more. Let's discuss this topic in detail, keeping it broad, yet Find out in simple terms, with lots of examples and step-by-step explanations.

Understanding polynomials

Before learning about factorization, let us first understand what polynomials are. A polynomial is a mathematical expression containing variables and coefficients, which are combined using addition, subtraction, and multiplication. Here is the general form of a polynomial:

P(x) = a n x n + a n-1 x n-1 + ... + a 1 x + a 0

In this expression:

  • a n , a n-1 , ..., a 0 are constants called coefficients.
  • x is the variable.
  • n is a non-negative integer, and it is the highest power of x that appears in the polynomial, known as the degree of the polynomial.

Concept of factorization

Factoring refers to the process of expressing a mathematical entity (such as a number or polynomial) as a product of its factors. In the context of polynomials, factoring involves expressing the polynomial as the product of two or more polynomials of lower degree These polynomials are called factors of the original polynomial.

An analogy

Think of factorization as breaking a number into its prime factors. For example, the number 18 can be factored into 2 × 3 × 3. Similarly, a polynomial can be factored into simpler polynomials.

Methods of factorization

There are several ways to factorize polynomials. Let us discuss them with examples:

1. Finding the greatest common factor (GCF)

The simplest method of factorization is to find the greatest common factor of the terms of a polynomial. Consider the polynomial:

P(x) = 12x 3 + 18x 2 + 6x

First find the GCF of the coefficients 12, 18, and 6, which is 6. Also, note that each term contains x. Therefore, the GCF is 6x.

We can factor out 6x from each term:

P(x) = 6x(2x 2 + 3x + 1)

Now, the polynomial is expressed as the product of 6x and (2x 2 + 3x + 1).

2. Factoring by grouping

Another technique used to factor polynomials, especially polynomials with four terms, is factoring by grouping. For example, consider:

Q(x) = x 3 + 3x 2 + 2x + 6

Start by grouping the words:

Q(x) = (x 3 + 3x 2 ) + (2x + 6)

Next, find the GCF for each group:

Q(x) = x 2 (x + 3) + 2(x + 3)

Note that (x + 3) is the same in both groups, so factor it out:

Q(x) = (x + 3)(x 2 + 2)

Thus, the polynomial is now divided as the product of (x + 3) and (x 2 + 2).

3. Factoring quadratic polynomials

The form of a quadratic polynomial is as follows:

ax 2 + bx + c

When factoring quadratics, we try to express them as the product of two binomials:

ax 2 + bx + c = (px + q)(rx + s)

Let's look at an example:

R(x) = x 2 + 5x + 6

We need two numbers whose product is 6 and sum is 5, such as 2 and 3. The factorization is as follows:

(x + 2)(x + 3)

Factorization is confirmed by multiplying:

(x + 2)(x + 3) = x 2 + 3x + 2x + 6 = x 2 + 5x + 6

4. Difference of squares

The difference of squares is a specific type of factorization. A polynomial that is the difference of two squares can be expressed as follows:

a 2 - b 2 = (a + b)(a - b)

Example:

S(x) = x 2 - 16

This is the difference of squares, since x 2 and 16 are perfect squares. Therefore, it can be divided as follows:

(x + 4)(x - 4)

5. Sum and difference of cubes

For cubes we have the sum and difference formulas:

a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) a 3 - b 3 = (a - b)(a 2 + ab + b 2 )

Consider a polynomial:

T(x) = x 3 - 8

This can be viewed as the difference of cubes:

x 3 - 2 3

Using the formula, this is factored as:

(x - 2)(x 2 + 2x + 4)

Practical example of polynomial factorization

Let's factor the polynomial completely:

U(x) = 2x 4 + 8x 3 + 6x 2
  1. Factor out the GCF:
    2. GCF = 2x 2
    U(x) = 2x 2 (x 2 + 4x + 3)
  2. Now, factor the quadratic inside the brackets:
    3. x 2 + 4x + 3 = (x + 1)(x + 3)
  3. Thus, the complete factorization becomes:
    4. U(x) = 2x 2 (x + 1)(x + 3)

Visualizing polynomial factorization

Let's see how polynomial factorization works with a simple example:

V(x) = x 2 - 5x + 6

This polynomial can be systematically factored into the following:

(x - 2)(x - 3)
x 2 -5x 6 (x – 2) (x – 3)

This visualization shows that factorization is equivalent to arranging objects into groups represented by a binomial.

Applications of factorization

Factorization has many applications in mathematics. Here are a few:

  • Simplifying fractions: Factoring can help simplify complex polynomial fractions, making them easier to calculate.
  • Solving a polynomial equation: Once factored, a polynomial equation can be easily solved by setting each factor equal to zero and solving for the variable.
  • Analyzing the graph: The discrete form of the polynomial can provide insights about the graph, such as the origin (where the graph intersects the x-axis).

Conclusion

Factoring is a fundamental aspect of polynomial algebra that allows complex expressions to be managed more easily. By breaking down polynomials into their component parts, we can solve equations, simplify terms, and understand mathematical relationships more clearly. The techniques covered through several examples demonstrate the process and utility of polynomial factorization in various contexts.


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Factorization of Polynomials

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