Algebraic manipulation: Class 9 Mathematics Notes, Short Questions and Numericals

HCF and LCM of polynomials, relation between HCF, LCM and the product, algebraic fractions in lowest terms, addition, subtraction, multiplication and division of fractions, and square roots of algebraic expressions.

Key points

Algebraic manipulation means using the rules of algebra to simplify or rewrite expressions. Almost every method in this chapter depends on factorization, so each polynomial is first written as a product of its irreducible factors. Then common parts can be identified and cancelled.

The highest common factor (HCF) of two or more polynomials is the polynomial of highest degree that divides each of them exactly. To find it, factorize each expression completely, then multiply the common factors, taking the lowest power of each. Numerical coefficients are handled in the same way: take their numerical HCF.

The lowest common multiple (LCM) is the polynomial of lowest degree that is exactly divisible by each given polynomial. To find it, factorize completely, then multiply every different factor that appears, taking the highest power of each. Include the numerical LCM of the coefficients.

Example: for x² − 4 = (x − 2)(x + 2) and x² + 5x + 6 = (x + 2)(x + 3), the HCF is x + 2. The LCM is (x − 2)(x + 2)(x + 3). Notice that the common factor x + 2 appears once in the LCM.

For two polynomials A and B: HCF × LCM = A × B. This is true for two expressions only, not for three or more. It can be used to find the LCM when the HCF and the two expressions are known: LCM = (A × B) / HCF. It also gives a quick check of an answer.

An algebraic fraction is a ratio of two polynomials, P/Q, with Q not zero. It is in lowest terms when P and Q have no common factor other than 1. To reduce it, factorize both numerator and denominator and cancel common factors. For (x² − 9)/(x² + 3x) we get (x − 3)(x + 3) / [x(x + 3)] = (x − 3)/x, for x not equal to 0 and −3.

Operations with algebraic fractions:

Multiplication: (P/Q) × (R/S) = PR / QS. Factorize first and cancel before multiplying.
Division: (P/Q) ÷ (R/S) = (P/Q) × (S/R). Invert the second fraction and multiply.
Addition and subtraction with the same denominator: P/Q ± R/Q = (P ± R)/Q.
With different denominators: find the LCM of the denominators, change each fraction to it, then add or subtract the numerators.

Example: 1/(x − 1) + 1/(x + 1). The LCM of the denominators is (x − 1)(x + 1) = x² − 1. Then the sum is [(x + 1) + (x − 1)] / (x² − 1) = 2x / (x² − 1). Always check whether the final fraction can be reduced further.

Take care with subtraction: when subtracting a fraction, put the whole numerator of the second fraction in brackets, so that every sign changes. For (x + 2)/3 − (x − 1)/2 the common denominator is 6, giving [2(x + 2) − 3(x − 1)] / 6 = (2x + 4 − 3x + 3)/6 = (7 − x)/6.

The square root of an algebraic expression is found when the expression is a perfect square. If the expression can be written as a square of a polynomial, then its square root is that polynomial. For example, x² + 6x + 9 = (x + 3)², so its square root is x + 3 (strictly |x + 3|, or ±(x + 3) if both roots are wanted).

To find the square root by factorization, factorize the expression into pairs of equal factors and take one factor from each pair. For 4x² − 12x + 9, we have (2x − 3)(2x − 3), so the root is 2x − 3. For a fraction, take the root of the numerator and denominator separately.

For expressions of higher degree, a long division method like that for numbers is used. Arrange terms in descending powers, find the root of the first term, subtract its square, bring down terms, double the root found so far for the trial divisor and repeat. A remainder of zero shows a perfect square.

Study tip: Factorize before doing anything else. Most marks in this chapter are lost by cancelling wrongly or by sign errors with brackets. Write each step on its own line, state the common denominator, and reduce the final answer to lowest terms. For HCF and LCM questions, use the product check to confirm your answer.

Definitions and flashcards

Say the meaning first, then open the card to check.

HCF

The polynomial of highest degree that divides each given polynomial exactly.

LCM

The polynomial of lowest degree that each given polynomial divides exactly.

Algebraic fraction

A ratio of two polynomials with a non-zero denominator.

Lowest terms

A fraction whose numerator and denominator have no common factor.

Common denominator

A denominator shared by fractions, usually the LCM of the old ones.

Perfect square expression

A polynomial that is the square of another polynomial.

Reciprocal

The fraction obtained by turning P/Q upside down to Q/P.

Irreducible factor

A factor that cannot be broken down any further.

Formula sheet

Name	Formula	What it means
HCF and LCM	HCF × LCM = A × B	Holds for two polynomials A and B.
Product of fractions	(P/Q) × (R/S) = PR / QS	Multiply numerators and denominators.
Quotient of fractions	(P/Q) ÷ (R/S) = PS / QR	Invert the divisor and multiply.
Sum of fractions	P/Q + R/S = (PS + QR) / QS	Add with a common denominator.
Perfect square	a² ± 2ab + b² = (a ± b)²	Its square root is a ± b.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Define the HCF of two polynomials. (2 marks)

The HCF of two polynomials is the polynomial of highest degree that divides both of them exactly, leaving no remainder. It is found from the common factors of their complete factorizations.

State the relation between HCF, LCM and the two given polynomials. (2 marks)

For two polynomials A and B, the product of their HCF and LCM equals the product of the polynomials: HCF × LCM = A × B. It is true for two expressions only.

Reduce (x² − 1)/(x² + x) to lowest terms. (2 marks)

Factorize: x² − 1 = (x − 1)(x + 1) and x² + x = x(x + 1). The common factor is x + 1. Cancelling gives (x − 1)/x.

Find the LCM of 6x²y and 9xy³. (3 marks)

Numerical LCM of 6 and 9 is 18. The highest power of x is x² and of y is y³. So the LCM is 18x²y³.

Simplify (x + 2)/3 − (x − 1)/2. (3 marks)

The LCM of 3 and 2 is 6. Rewrite: [2(x + 2) − 3(x − 1)] / 6. Expand: (2x + 4 − 3x + 3)/6. Combine: (7 − x)/6.

Explain how to find the square root of 4x² − 12x + 9 by factorization and check your answer. (5 marks)

Look for a perfect square. The first term 4x² = (2x)² and the last term 9 = 3². The middle term 12x = 2 × 2x × 3, so the expression is (2x − 3)². Its square root is 2x − 3. Check: (2x − 3)² = 4x² − 12x + 9, which matches the given expression.

Numericals with worked steps

Find the HCF and LCM of x² − 4 and x² + 5x + 6.

x² − 4 = (x − 2)(x + 2).
x² + 5x + 6 = (x + 2)(x + 3).
Common factor: x + 2, so HCF = x + 2.
LCM takes all different factors: (x − 2)(x + 2)(x + 3).

Answer: HCF = x + 2 and LCM = (x − 2)(x + 2)(x + 3)

Simplify (x² − 9)/(x² + 3x) in lowest terms.

Numerator: x² − 9 = (x − 3)(x + 3).
Denominator: x² + 3x = x(x + 3).
Cancel the common factor x + 3.

Answer: (x − 3)/x

Simplify 1/(x − 1) + 1/(x + 1).

Common denominator: (x − 1)(x + 1) = x² − 1.
First fraction becomes (x + 1)/(x² − 1), second becomes (x − 1)/(x² − 1).
Add numerators: (x + 1) + (x − 1) = 2x.

Answer: 2x/(x² − 1)

Find the square root of x² + 6x + 9 and of 4x² − 12x + 9.

x² + 6x + 9 = x² + 2(x)(3) + 3² = (x + 3)².
So its root is x + 3.
4x² − 12x + 9 = (2x)² − 2(2x)(3) + 3² = (2x − 3)².
So its root is 2x − 3.

Answer: x + 3 and 2x − 3

Verify HCF × LCM = A × B for A = x² − 4 and B = x² + 5x + 6.

HCF × LCM = (x + 2) × (x − 2)(x + 2)(x + 3).
A × B = (x − 2)(x + 2) × (x + 2)(x + 3).
Both are (x − 2)(x + 2)²(x + 3), so they are equal.

Answer: Verified: both sides equal (x − 2)(x + 2)²(x + 3)

Common mistakes

Cancelling terms instead of factors. In (x + 3)/3 you cannot cancel the 3s, but in 3(x + 1)/3 you can.
Taking the lowest power of each factor for the LCM; the LCM uses the highest power, and the HCF uses the lowest.
Forgetting to put brackets around the second numerator when subtracting fractions, so signs are not changed.
Adding fractions by adding numerators and denominators separately, for example 1/a + 1/b = 2/(a + b).
Applying HCF × LCM = product to three polynomials, where it does not hold.

Practise this chapter

Practice questions and timed test Printable worksheet