Highest common factor (H.C.F)

H.C.F by Factorization Method:

Finding the HCF of algebraic expressions by factorization involves factoring each expression into its irreducible factors, and then identifying the highest common factors.

Here's an example to illustrate the process:

Example: Find the HCF of $18x^2y^3$ and $12xy^2$.

Solution:

Step 1: Factorize each expression.

$18x^2y^3 = 2 \times 3^2 \times x^2 \times y^3$

$12xy^2 = 2^2 \times 3 \times x \times y^2$

Step 2: Identify the highest common factors.

The common factors are $2, 3, x,$ and $y^2$. The highest common factor of $2$ and $2^2$ is $2$. The highest common factor of $3$ and $3^2$ is $3$, the highest common factor of $x$ and $x^2$ is $x$, and the highest common factor of $y^2$ and $y^3$ is $y^2$.

Step 3: Multiply the common factors with their lowest powers to get the HCF.

$\text{HCF} = 2 \times 3 \times x \times y^2 = 6xy^2$

Therefore, the H.C.F of $18x^2y^3$ and $12xy^2$ is $6xy^2$.

Example: Find the H.C.F of the following expression using the factorization method: $(x+1)^2, \;\;x^2 - 1, \;\;x^2 + 4x + 3$.

Solution:

Step 1: Factorize each expression.

$(x+1)^2 = (x+1)(x+1)$ $x^2 - 1 = (x+1)(x-1)$ $x^2 + 4x + 3 = (x+1)(x+3)$

Step 2: Identify the highest common factors.

The common factor is only $(x+1)$. And hence it is the highest common factor.

Step 3: Multiply the common factors with their lowest powers to get the H.C.F.

Since $(x+1)$ is the only common factor, therefore, it is indeed the H.C.F.

Hence the H.C.F of $(x+1)^2$, $x^2 - 1$ and $x^2 + 4x + 3$ is $(x+1)$.

H.C.F by Division Method: Finding the HCF of algebraic expressions by division method follows a division. The procedure is explained with the help of the following examples.

Example: Find the H.C.F of $2x^3+7x^2 + 4x - 4$ and $2x^3+9x^2 + 11x + 2$ using Division Method.

Solution:

Step 1: Divide the highest-order expression by the other. If they are of the same order then choice does not matter. Divide any expression by the other.

Here since both expression has the same order as 3, therefore;

Step 2: Now divide the divisor by the remainder.

Take $-2$ common from $-2x-4$ and omit it, i.e. new remainder is $x+2$.

Step 3: Repeat Step 2 until the remainder is zero.

Step 4: The last divisor is the H.C.F.

Since the last divisor is $x+2$.

Hence, $x+2$ is the H.C.F of $2x^3+7x^2 + 4x - 4$ and $2x^3+9x^2 + 11x + 2$

Example: Find the H.C.F of the following expression using division method: $(x+1)^2, \;\;x^2 - 1, \;\;x^2 + 4x + 3$.

Solution: For the H.C.F of three expressions, firstly find the H.C.F of 2 expressions, then find the H.C.F of the previous H.C.F with the last expression.

$(x+1)^2 = x^2+2x+1$, since all three expressions have the same order, therefore we can choose any two expressions first.

Step 1: Divide the highest-order expression by the other. If they are of the same order then choice does not matter. Divide any expression by the other.

Divide any of the above expressions by the other,

Take $2$ common from $2x+2$ and omit it, i.e. new remainder is $x+1$.

Step 2: Now divide the divisor by the remainder.

Step 3: Since the remainder is already zero, so Step 3 is completed.

Step 4: Since the last divisor is $x+1$, therefore the H.C.F of $(x+1)^2$ and $x^2+4x + 3$ is $x+1$.

Now find the H.C.F of $x+1$ and $x^2-1$.

Step 1: Divide the highest-order expression by the other. If they are of the same order then choice does not matter. Divide any expression by the other.

Step 2: Since the remainder is already zero, steps 2 and 3 are completed.

Step 4: Since the last divisor is $x+1$, therefore the H.C.F of $x+1$ and $x^2-1$ is $x+1$.

Hence the H.C.F of $(x+1)^2, \;\;x^2 - 1, \;\;x^2 + 4x + 3$. is $(x+1)$.