Common factor
When a photo-editing app shrinks a picture, it does not recalculate each pixel on its own: it applies the same scale factor to all of them at once. Taking out a common factor is that same idea in algebra: finding what all the terms have in common so that we can write them as a single product.
Common factor of a monomial
It is the distributive property in reverse: if all the terms share a factor, you take it outside, multiplying a bracket.
$$12x^4 - 18x^3 + 6x^2 = 6x^2\,(2x^2 - 3x + 1)$$The common factor is built in two independent steps: the numerical part (the H.C.F. of the coefficients) and the literal part (each letter that appears in every term, raised to the lowest power with which it appears).
Factorise out the highest common factor of the polynomial $P(x) = 12x^4 - 18x^3 + 6x^2$.
Factorise out the highest common factor of the polynomial $P(x) = 12x^4 - 18x^3 + 6x^2$.
Factorise out the common factor in the following algebraic expression with two variables: $15x^3y^2 - 20x^2y^3 + 5x^2y^2$.
Factorise out the common factor in the following algebraic expression with two variables: $15x^3y^2 - 20x^2y^3 + 5x^2y^2$.
A student has factorised out a common factor in the expression $4x^3 - 8x^2 + 4x$ giving the result $4x(x^2 - 2x)$. Identify the error made and write down the correct…
A student has factorised out a common factor in the expression $4x^3 - 8x^2 + 4x$ giving the result $4x(x^2 - 2x)$. Identify the error made and write down the correct factorisation.
Common factor with several variables and three terms
The idea does not change with more letters: each variable contributes to the common factor the lowest power with which it appears in all the terms, even if there are three or more.
Factorise out the common factor in the following expression with three variables: $7a^2bc - 14ab^2c + 21abc^2$.
Factorise out the common factor in the following expression with three variables: $7a^2bc - 14ab^2c + 21abc^2$.
Factorise out the common factor in the expression $M(x, y) = 18x^2y^4 - 24x^3y^3 + 30x^4y^2$.
Factorise out the common factor in the expression $M(x, y) = 18x^2y^4 - 24x^3y^3 + 30x^4y^2$.
Express the polynomial $R(t) = 16t^5 - 24t^4 + 32t^3$ by taking out the highest common factor.
Express the polynomial $R(t) = 16t^5 - 24t^4 + 32t^3$ by taking out the highest common factor.
Common factor with a repeated bracket
Sometimes the common factor is not a monomial but a whole bracket that is repeated in several terms. It is treated in exactly the same way: it comes out, multiplying.
$$x(x-3)^2 - 4(x-3)^2 + x^2(x-3)^2 = (x-3)^2\,(x - 4 + x^2)$$Factorise out the common factor in the expression $E(x) = x(x-3)^2 - 4(x-3)^2 + x^2(x-3)^2$.
Factorise out the common factor in the expression $E(x) = x(x-3)^2 - 4(x-3)^2 + x^2(x-3)^2$.
Common factor by grouping
If a polynomial with four terms has no factor common to all of them, it can sometimes be grouped in pairs: take out the common factor of each pair separately, and discover that what is left inside the brackets is the same in both. That bracket then becomes the new common factor.
Perform a factorisation by grouping terms (taking out common factors twice) in the polynomial $P(x, y) = x^3 - 2x^2y + 3x - 6y$.
Perform a factorisation by grouping terms (taking out common factors twice) in the polynomial $P(x, y) = x^3 - 2x^2y + 3x - 6y$.
Factorise the following expression by grouping terms: $5a^2b + 15a^2 - 2b - 6$.
Factorise the following expression by grouping terms: $5a^2b + 15a^2 - 2b - 6$.
Factorise the three-variable polynomial $Q(a, b, c) = a^2b - a^2c + ab^2 - ac^2 - b^2c + bc^2$ by grouping terms. Group in pairs and find a common binomial factor.
Factorise the three-variable polynomial $Q(a, b, c) = a^2b - a^2c + ab^2 - ac^2 - b^2c + bc^2$ by grouping terms. Group in pairs and find a common binomial factor.
Factorising completely: common factor + special products
Factorising "completely" means continuing to take out factors until you cannot go any further: first the common factor, and if what is left inside the bracket is a special product, you factorise that too.
Factorise out the common factor in the polynomial $P(x) = 5x^3 - 30x^2 + 45x$ and check whether the resulting quadratic factor is a standard algebraic identity. Write…
Factorise out the common factor in the polynomial $P(x) = 5x^3 - 30x^2 + 45x$ and check whether the resulting quadratic factor is a standard algebraic identity. Write down the complete factorisation.
The volume of a right rectangular prism is given by the polynomial $V(x) = 3x^3 + 12x^2 + 12x$. Factorise out the common factor and apply standard algebraic identities…
The volume of a right rectangular prism is given by the polynomial $V(x) = 3x^3 + 12x^2 + 12x$. Factorise out the common factor and apply standard algebraic identities to express the volume as the product of three linear polynomials representing the dimensions of the prism.
Fully factorise the polynomial $P(x) = x^{n+3} - 4x^{n+2} + 4x^{n+1}$ by first taking out the suitable common factor, assuming that $n$ is a natural number.
Fully factorise the polynomial $P(x) = x^{n+3} - 4x^{n+2} + 4x^{n+1}$ by first taking out the suitable common factor, assuming that $n$ is a natural number.
What it is for: roots and algebraic fractions
Factorising is not an isolated exercise: it is the tool you will soon use for two very different things. First, to find the values of $x$ that make a polynomial equal to $0$: if $P(x) = 5x(x-3)^2$, you just need to see which factor becomes zero, with no need for complicated formulae. Second, to simplify algebraic fractions: you can only cancel what appears as a complete factor in both numerator and denominator, never a single term inside a sum.
This means that the later sections on roots of a polynomial and on algebraic fractions depend directly on your being able to take out a common factor without hesitating: the more automatic it is now, the less work you will have there.
In an electrical circuit, the power dissipated by three resistors in series is expressed as $P(I) = 12I^3 - 18I^2 + 6I$. Factorise out the common factor and find the…
In an electrical circuit, the power dissipated by three resistors in series is expressed as $P(I) = 12I^3 - 18I^2 + 6I$. Factorise out the common factor and find the current values $I$ for which the power is zero.
An architect designs a structure whose supported weight as a function of a structural variable $x$ is $W(x) = \frac{1}{2}x^4 - 2x^3 + 2x^2$. Factorise out common factors…
An architect designs a structure whose supported weight as a function of a structural variable $x$ is $W(x) = \frac{1}{2}x^4 - 2x^3 + 2x^2$. Factorise out common factors and fully factorise the polynomial. For which values of $x > 0$ does the structure support no weight?