Polynomials - Algebraic identities, remainder theorem, factor theorem, and factorization

Hello dear students! Welcome to this comprehensive guide on Polynomials.

If you have ever played with LEGO bricks, you already understand polynomials! Just as you build complex structures by snapping together basic bricks, in algebra, we build complex expressions by combining numbers, variables, and exponents.

In Class 9, this chapter is one of the most important scoring topics. It also forms the backbone for Class 10 Board exams and higher mathematics. Let us break down the core concepts step-by-step with simple logic, real-world analogies, and clear examples!

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1. Quick Recap: What is a Polynomial?

A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents.

$$\text{General Form: } p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

Variables: Symbols like $x, y, z$ that can take different values.

Coefficients: Real numbers attached to the variables (e.g., in $5x^2$, $5$ is the coefficient).

Exponent Rule: The power of the variable must be a whole number ($0, 1, 2, 3, \dots$).

$3x^2 + 2x + 1$ $\rightarrow$ Polynomial

$x^{-2} + 5$ or $\sqrt{x} + 3$ $\rightarrow$ NOT a Polynomial (powers are negative or fractional).

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2. The Remainder Theorem

Real-World Analogy: Division without Doing the Long Work

Imagine you want to know if 100 chocolates can be equally shared among 7 friends, and if not, how many will be left over. You could perform full long division, or you could use a quick mathematical trick to find *just the remainder*.

The Remainder Theorem does exactly this for polynomials! It gives you the remainder of a division without performing tedious long division.

Statement of the Theorem

Let $p(x)$ be any polynomial of degree greater than or equal to $1$, and let $a$ be any real number. If $p(x)$ is divided by the linear polynomial $(x - a)$, then the remainder is $p(a)$.

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Traditional Long Division: Remainder Theorem Shortcut:

Polynomial p(x) ÷ (x - a) ======> Step 1: Set divisor x - a = 0 => x = a

[Takes 5-10 minutes] Step 2: Calculate p(a) directly!

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Step-by-Step Procedure

Take the linear divisor and set it to zero: $x - a = 0 \implies x = a$.

Substitute $x = a$ into the polynomial $p(x)$.

The resulting value $p(a)$ is your Remainder.

Solved Example:

Find the remainder when $p(x) = x^3 - 3x^2 + 4x + 5$ is divided by $(x - 2)$.

Step 1: Find the zero of the divisor $(x - 2)$.

$$x - 2 = 0 \implies x = 2$$

Step 2: Substitute $x = 2$ into $p(x)$.

$$p(2) = (2)^3 - 3(2)^2 + 4(2) + 5$$

$$p(2) = 8 - 3(4) + 8 + 5$$

$$p(2) = 8 - 12 + 8 + 5 = 9$$

Answer: The remainder is $9$.

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3. The Factor Theorem

Real-World Analogy: The Perfect Key

Think of a lock and a key. If a key turns smoothly with zero resistance (zero remainder), it is the correct key for that lock.

In mathematics, if dividing $p(x)$ by $(x - a)$ leaves a remainder of $0$, then $(x - a)$ is a factor (a perfect key) of $p(x)$.

Statement of the Theorem

For a polynomial $p(x)$:

1. If $p(a) = 0$, then $(x - a)$ is a factor of $p(x)$.

2. Conversely, if $(x - a)$ is a factor of $p(x)$, then $p(a) = 0$.

Solved Example:

Examine whether $(x + 2)$ is a factor of $p(x) = x^3 + 3x^2 + 5x + 6$.

Step 1: Find the zero of $(x + 2)$.

$$x + 2 = 0 \implies x = -2$$

Step 2: Substitute $x = -2$ into $p(x)$.

$$p(-2) = (-2)^3 + 3(-2)^2 + 5(-2) + 6$$

$$p(-2) = -8 + 3(4) - 10 + 6$$

$$p(-2) = -8 + 12 - 10 + 6 = 0$$

Step 3: Conclusion.

Since $p(-2) = 0$, by the Factor Theorem, $(x + 2)$ is indeed a factor of $p(x)$.

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4. Factorization of Polynomials

Factorization means breaking down a polynomial into a product of simpler polynomials.

Method 1: Splitting the Middle Term (For Quadratic Polynomials: $ax^2 + bx + c$)

To factorize $ax^2 + bx + c$, we need to find two numbers $p$ and $q$ such that:

$p + q = b$ (the middle coefficient)

$p \times q = a \times c$ (product of first and last coefficients)

Solved Example: Factorize $6x^2 + 17x + 5$

Here, $a = 6, b = 17, c = 5$.

Product $a \times c = 6 \times 5 = 30$.

We need two numbers that multiply to $30$ and add up to $17$. The numbers are $15$ and $2$ (since $15 \times 2 = 30$ and $15 + 2 = 17$).

Split the middle term $17x$ into $15x + 2x$:

$$6x^2 + 15x + 2x + 5$$

Group terms in pairs and take out common factors:

$$3x(2x + 5) + 1(2x + 5)$$

Factor out the common binomial $(2x + 5)$:

$$(2x + 5)(3x + 1)$$

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Method 2: Trial Method using Factor Theorem (For Cubic Polynomials: $ax^3 + bx^2 + cx + d$)

A cubic polynomial has at most 3 linear factors.

Steps:

Find factors of the constant term $d$.

Test these factors using hit-and-trial until you find one value $a$ such that $p(a) = 0$. This gives your first factor $(x - a)$.

Divide $p(x)$ by $(x - a)$ to get a quadratic polynomial.

Factorize the quadratic polynomial using the splitting middle term method.

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5. Algebraic Identities (Class 9 Master List)

Identities are algebraic equations that are true for all values of the variables. Think of them as ultimate mathematical shortcuts!

Here are the 8 fundamental identities you MUST memorize for Class 9:

No.	Algebraic Identity
1	$(x + y)^2 = x^2 + 2xy + y^2$
2	$(x - y)^2 = x^2 - 2xy + y^2$
3	$x^2 - y^2 = (x + y)(x - y)$
4	$(x + a)(x + b) = x^2 + (a + b)x + ab$
5	$(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$
6	$(x + y)^3 = x^3 + y^3 + 3xy(x + y) = x^3 + 3x^2y + 3xy^2 + y^3$
7	$(x - y)^3 = x^3 - y^3 - 3xy(x - y) = x^3 - 3x^2y + 3xy^2 - y^3$
8	$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$

Teacher's Golden Rule for Identity 8:

If $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$. This shortens huge calculations instantly!

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6. Master Practice Corner (3 Board-Style Questions)

Let us test your understanding with these step-by-step practice problems!

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Practice Question 1: Finding an Unknown Constant

Question: Find the value of $k$ if $(x - 1)$ is a factor of $p(x) = 2x^2 + kx + \sqrt{2}$.

Solution:

Apply the Factor Theorem:

Since $(x - 1)$ is a factor of $p(x)$, the zero of $(x - 1)$ must make $p(x) = 0$.

$$x - 1 = 0 \implies x = 1$$

Therefore, $p(1) = 0$.

Substitute $x = 1$ into $p(x)$:

$$p(1) = 2(1)^2 + k(1) + \sqrt{2} = 0$$

$$2(1) + k + \sqrt{2} = 0$$

$$2 + k + \sqrt{2} = 0$$

Solve for $k$:

$$k = -2 - \sqrt{2} = -(2 + \sqrt{2})$$

Final Answer: $k = -(2 + \sqrt{2})$

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Practice Question 2: Factorizing a Cubic Polynomial

Question: Factorize $p(x) = x^3 - 23x^2 + 142x - 120$.

Solution:

Step 1: Find the first factor by Trial Method.

Look at factors of the constant term $-120$: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5 \dots$

Let's test $x = 1$:

$$p(1) = (1)^3 - 23(1)^2 + 142(1) - 120$$

$$p(1) = 1 - 23 + 142 - 120$$

$$p(1) = 143 - 143 = 0$$

Since $p(1) = 0$, by Factor Theorem, $(x - 1)$ is a factor.

Step 2: Divide $p(x)$ by $(x - 1)$ to find the remaining quadratic factor.

Using long division or term manipulation:

$$x^3 - 23x^2 + 142x - 120 = x^3 - x^2 - 22x^2 + 22x + 120x - 120$$

$$= x^2(x - 1) - 22x(x - 1) + 120(x - 1)$$

$$= (x - 1)(x^2 - 22x + 120)$$

Step 3: Factorize the quadratic polynomial $x^2 - 22x + 120$.

We need two numbers that multiply to $+120$ and add up to $-22$. The numbers are $-12$ and $-10$.

$$x^2 - 12x - 10x + 120$$

$$= x(x - 12) - 10(x - 12)$$

$$= (x - 12)(x - 10)$$

Step 4: Combine all factors.

$$p(x) = (x - 1)(x - 10)(x - 12)$$

Final Answer: $(x - 1)(x - 10)(x - 12)$

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Practice Question 3: Smart Evaluation using Identities

Question: Evaluate the following without direct expansion/multiplication:

$105 \times 106$

$(-12)^3 + (7)^3 + (5)^3$

Solution:

Part 1: Evaluate $105 \times 106$

Rewrite using Identity 4: $(x + a)(x + b) = x^2 + (a + b)x + ab$

$$105 \times 106 = (100 + 5)(100 + 6)$$

Here, $x = 100, a = 5, b = 6$.

$$(100 + 5)(100 + 6) = (100)^2 + (5 + 6)(100) + (5 \times 6)$$

$$= 10000 + 11(100) + 30$$

$$= 10000 + 1100 + 30 = 11130$$

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Part 2: Evaluate $(-12)^3 + (7)^3 + (5)^3$

Let $x = -12$, $y = 7$, and $z = 5$.

First, test the sum of $x, y, z$:

$$x + y + z = -12 + 7 + 5 = 0$$

Since $x + y + z = 0$, we can use the conditional identity:

$$x^3 + y^3 + z^3 = 3xyz$$

Substitute the values:

$$(-12)^3 + (7)^3 + (5)^3 = 3(-12)(7)(5)$$

$$= 3 \times (-12) \times 35$$

$$= -36 \times 35 = -1260$$

Final Answer:

$11130$

$-1260$

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🌟 Teacher's Tip for Success

Always check the degree of your polynomial before applying theorems.

Watch out for signs! A common mistake is forgetting that $x - a = 0 \implies x = +a$, while $x + a = 0 \implies x = -a$.

Practice rewriting numbers into familiar identity formats (like writing $99$ as $100 - 1$).

Keep practicing these concepts, write down the identities twice daily, and you will ace this chapter with full confidence! Happy Learning!