Exercises: Common factor

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 $-3x$ from the polynomial $Q(x) = -3x^3 + 12x^2 - 9x$.

The perimeter of an enclosure is given by the expression $P = 8x^2 + 12x$. Write this expression as the product of two factors by factorising out the common factor.

Factorise out the common factor in the following expression with three variables: $7a^2bc - 14ab^2c + 21abc^2$.

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 factorisation.

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.

At a municipal sports centre in Seville, the monthly maintenance cost of the padel and tennis courts depends on the number $x$ of tournaments held and is given by the expression $C(x) = 15x^2 + 45x$ euros.

1. a) Before doing any algebraic work, make a mental estimate: if $x = 10$ tournaments are organized in May, will the cost be closer to $1.500$ euros or $2.000$ euros? Briefly justify your approximation without calculating the exact value yet.
2. b) Factor out the greatest common monomial factor of $C(x)$ to transform the expression into a product of the form $k \cdot x \cdot (x + a)$.
3. c) Calculate the exact cost by substituting $x = 10$ into the factored expression you obtained, and check whether your initial estimate was reasonable.

While simplifying a formula in class, Irene tries to factor out a common factor in the polynomial $P(x) = 6x^4 - 3x^2$ and writes on the board:

When her teacher points out that the factorization is not correct, Irene protests: "The term $3x^2$ was exactly the second term of the polynomial, so by taking it out, there is nothing left inside the parentheses."

1. a) Explain in detail what Irene's mistake is and what term should be in its place inside the parentheses, relying on the distributive property to justify your answer.
2. b) Write the complete and correct factorization of $P(x)$.

Perform a factorisation by grouping terms (taking out common factors twice) in the polynomial $P(x, y) = x^3 - 2x^2y + 3x - 6y$.

The total surface area of a solid cylinder of radius $r$ and height $h$ is given by the expression $A = 2\pi r^2 + 2\pi rh$. Factorise out the common factor to obtain a more compact formula and calculate the exact area if $r = 5$ cm and $h = 12$ cm using the factorised expression.

A company's production costs based on the number of units manufactured $x$ are given by $C(x) = \frac{3}{4}x^3 - \frac{9}{8}x^2 + \frac{15}{4}x$. Factorise out the highest common factor with fractional coefficients so that the remaining polynomial has integer coefficients.

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 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 to express the volume as the product of three linear polynomials representing the dimensions of the prism.

Factorise the following expression by grouping terms: $5a^2b + 15a^2 - 2b - 6$.

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.

During a review session, the teacher erases the board and leaves only the clues to a mystery polynomial $P(x)$ of degree 3:

• The leading term of $P(x)$ has coefficient $4$.

• When its greatest common monomial factor, which is $4x$, is factored out, a quadratic trinomial remains inside parentheses.

• This trinomial in parentheses is exactly the expansion of the square of a difference: $(x - 3)^2$.

1. a) Reconstruct the original polynomial $P(x)$ by writing it in its expanded and simplified form ($a x^3 + b x^2 + c x$). Show the intermediate steps by first expanding the notable product and then applying the distributive property.
2. b) Without performing divisions or additional calculations, deduce from its factored expression what the real roots of $P(x)$ are, and explain how you obtained each one.

In an irrigation community in Campo de Cartagena (Murcia), a pumping station supplies water to two agricultural sectors according to the hours $t$ of continuous operation ($0 \le t \le 10$). The daily volumes supplied (in cubic metres) are modelled by:

• Citrus sector: $V_1(t) = 8t^2 + 24t$

• Vegetable sector: $V_2(t) = 4t^3 + 12t^2$

1. a) Write the expression for the total volume supplied $V_T(t) = V_1(t) + V_2(t)$ in simplified form, and factor it by extracting the greatest possible common monomial factor.
2. b) To assess energy efficiency, the technicians study the ratio between both consumptions using the algebraic fraction $\frac{V_2(t)}{V_1(t)}$. Factor the numerator and the denominator, simplify the fraction by cancelling out common complete factors, and calculate for how many hours of irrigation ($t$) the volume consumed by vegetables is three times that of citrus fruits.

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.

The figure shows a rectangle divided into four regions of areas $A_1 = x^2$, $A_2 = 3x$, $A_3 = 2x$ and $A_4 = 6$. Express the total area as a polynomial, factorise by grouping terms, and deduce the expressions for the base and height of the overall rectangle.

Find the value of the parameters $a$ and $b$ knowing that after factorising out $2x^2$ from the polynomial $P(x) = 2x^4 + ax^3 + bx^2$, the resulting polynomial is the square of a binomial, the coefficient of $x^3$ is negative and $P(1) = 18$ holds.

Factorise the following complex algebraic expression by repeatedly taking out common factors: $M(x, y) = x^2(x-y) - y^2(x-y) - x(x-y)^2 + y(x-y)^2$.

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?

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.

In the study of beam deflection, the polynomial $D(x) = kx^5 - 3kx^4 + 3kx^3 - kx^2$ appears, where $k$ is a non-zero constant. Factorise out the common factor and factorise the remaining polynomial knowing that $x=1$ is a multiple root.

Prove algebraically that the expression $E = (2n+1)^2 - (2n-1)^2$ is always a multiple of $8$ for any integer $n$, by expanding the squares and factorising out a common factor from the result.

To completely factor the polynomial $P(x) = 3x^3 - 15x^2 - 12x + 60$ and find its roots, two students propose different methods:

• Mateo claims that the safest way is to directly use Ruffini's rule, testing the integer divisors of the constant term $60$.

• Alba argues that it is much faster and more elegant to first factor out the numerical common factor $3$, group the four resulting terms in pairs (factoring by grouping), and then apply a notable product.

1. a) Fully carry out Alba's procedure: factor out the numerical common factor, perform the double extraction by grouping terms, and decompose the final difference of squares to obtain the three degree-1 factors.
2. b) Compare both methods and critically justify why Alba's proposal is clearly more advantageous than Mateo's in this case: how many divisors of $60$ would Mateo have to test at most, and what calculation risks does his method carry?

According to technical data from Red Eléctrica de España (REE), the estimated power generation curve (in kilowatts, kW) of a small photovoltaic solar plant in Extremadura during a sunny day is modelled by the polynomial:

where $t$ represents the time in hours elapsed since sunrise at 6:00 (that is, $t = 0$ is 6:00 and $t = 12$ is 18:00 at sunset).

1. a) Factor out the greatest possible common monomial factor from the expression $P(t)$.
2. b) Verify that the trinomial remaining inside the parentheses is a perfect square trinomial (notable product) and write the complete factorization of the polynomial $P(t)$.
3. c) From the previous factorization, and without solving any quadratic equation, deduce at what times of day the power generated is exactly $0\text{ kW}$.
4. d) In mathematics, if a factor is squared, such as $(t - a)^2$, $a$ is said to be a "double root". Explain why it makes physical sense that at 18:00 ($t = 12$) there is a double root (the curve smoothly touches zero without crossing into negative power values) rather than a sharp crossing through the axis.