Standard algebraic identities

Aoife wants to design a square rug for her bedroom. The side of the rug measures $(x + 4)$ metres.

1. a) Write the expression for the area of the rug as the square of a binomial.
2. b) Expand the binomial to obtain the polynomial representing the area.

Expand the following expressions using algebraic identities to convert them into an ordered polynomial:

1. a) $(3x + 5)^2$

1. b) $(2x - 7y)^2$

1. c) $(x^2 + 4) \times (x^2 - 4)$

1. d) $(\frac{x}{2} - 6)^2$

A very common mistake when starting with algebraic identities is thinking that the square of a sum is simply the sum of the squares. For example:

Is it true that $(x + 3)^2 = x^2 + 3^2$?

To check this, calculate the numerical value of both expressions for $x = 2$ and explain whether the results are equal or different.

In a 2nd Year exam, Aoife solved the expansion of the expression $(2x - 3)^2$ by directly writing: $4x^2 - 9$. Her teacher pointed out that the answer is wrong because it contains two conceptual errors in that single step.

1. a) Identify the two errors made by Aoife (one regarding the sign of the last term and another regarding the missing terms in the expansion).
2. b) Apply the distributive property by multiplying $(2x - 3) \times (2x - 3)$ term by term to clearly explain to her where each element comes from, and write the final correct and simplified expression.

To cover the floor of a square stand at the Dublin Book Festival, a tarpaulin with a side length of $39\text{ m}$ needs to be ordered.

1. a) Before calculating the exact area, make a mental estimate by rounding the side length to the nearest ten ($40\text{ m}$). Reason whether the actual area of the stand will be greater than or less than your estimate.
2. b) Express the number $39$ as the subtraction $(40 - 1)$ and calculate the exact area by applying the remarkable identity of the square of a difference, indicating the three terms resulting from the expansion.
3. c) Subtract the calculated exact area from your initial estimate and check whether the difference matches what you reasoned in section a).

Fill in the blanks so that the following equations are correct:

1. a) $(x + 4)^2 = x^2 + \dots x + 16$

1. b) $(a - \dots)^2 = a^2 - 6a + 9$

1. c) $(y + 5)(y - 5) = y^2 - \dots$

Match each algebraic identity product in your exercise book with its corresponding expanded expression:

• A. $(x + 2)^2$

• B. $(x - 2)^2$

• C. $(x + 2)(x - 2)$

• $x^2 - 4$

• $x^2 + 4x + 4$

• $x^2 - 4x + 4$

To mentally calculate the result of the operation $105^2 - 95^2$ without using a calculator, Cian and Aoife propose two different strategies:

• Cian's method: Perform the two traditional multiplications separately, $105 \times 105$ and $95 \times 95$, and then subtract the second result from the first.

• Aoife's method: Identify that the expression is a difference of squares of the form $a^2 - b^2$, transform it into the product of a sum and a difference, and perform the resulting operations.

1. a) Apply the method proposed by Aoife, expanding the numerical calculation step by step until obtaining the final value.
2. b) Compare both methods and justify with reasons why Aoife's proposal is much more advantageous, analyzing the difficulty of the intermediate operations for each path.

Simplify the following algebraic expression as much as possible by expanding using algebraic identities first and then collecting like terms:

$E = (2x + 1)^2 - (2x - 1)^2 + (x + 1) \times (x - 1)$

Aoife and Seán are doing their maths homework and have made some common mistakes. Identify which of the following equations is incorrect and correct it in your exercise book, explaining why it is wrong:

1. a) $(x + 4)^2 = x^2 + 16$

1. b) $(x - 5) \times (x + 5) = x^2 - 25$

1. c) $(2a - 3)^2 = 4a^2 - 12a + 9$

In a debate in Mathematics class, Cian states: "The square of the sum of two numbers, $(a + b)^2$, always gives a result greater than the sum of their squares, $a^2 + b^2$, because when expanding it we always add the term $2 \times a \times b$".

1. a) Expand $(a + b)^2$ and calculate the algebraic difference $(a + b)^2 - (a^2 + b^2)$ in simplified form. What algebraic term completely determines whether the result is greater, less, or equal?
2. b) Prove that Cian's statement is false by providing a numerical counterexample with integers where $(a + b)^2 < a^2 + b^2$ holds. Next, investigate whether it is possible to find two non-zero numbers such that $(a + b)^2 = a^2 + b^2$, and justify your conclusion.
3. c) State precisely what mathematical condition the signs of $a$ and $b$ must satisfy for $(a + b)^2 > a^2 + b^2$, when it happens that $(a + b)^2 = a^2 + b^2$, and when it happens that $(a + b)^2 < a^2 + b^2$.

A local council wants to design a new square town square. In the centre there will be a landscaped garden area, also square, surrounded by a paved path of constant width.

If the total side length of the square is $(3x + 2)$ metres and the side length of the central garden is $(3x - 2)$ metres, calculate the simplified polynomial that represents the area intended solely for the paved path.

Look at the following figure representing a square with side length $(x + 3)$ from which an inner square of side length $(x - 1)$ has been cut out. Use algebraic identities to write a simplified expression representing the orange shaded area.

Find the area of the shaded region in the following figure. Express the result as a simplified polynomial in terms of $x$. The side of the outer square measures $2x+3$ and the side of the inner white square measures $2x-3$.

Simplify the following algebraic fractions using algebraic identities to factorise the numerator and the denominator:

1. a) $\frac{x^2 - 25}{x^2 + 10x + 25}$

1. b) $\frac{9x^2 - 12x + 4}{9x^2 - 4}$

Prove algebraically that the difference between the squares of two consecutive integers is always equal to the sum of these numbers.

(Hint: call the first number $x$ and the next one $x+1$).

Evaluate the numerical value of the expression $E = (3x + 4y)^2 - (3x - 4y)^2$ for the values $x = \frac{1}{3}$ and $y = 5$.

Tip: Simplify the algebraic expression using algebraic identities before substituting the values.

Copy and complete in your exercise book the missing terms so that these expressions form a perfect square trinomial or a difference of two squares:

1. a) $x^2 + 10x + \_\_\_\_ = (x + \_\_\_\_)^2$

1. b) $25a^2 - \_\_\_\_ + 1 = (\_\_\_\_ - 1)^2$

1. c) $16x^2 - \_\_\_\_ = (\_\_\_\_ + 3y) \times (\_\_\_\_ - 3y)$

Find the missing term in each expression so that the trinomial is the result of the square of a sum or a difference (a perfect square). Then, write the corresponding power:

1. a) $16x^4 + \dots + 9y^2$

1. b) $\frac{1}{4}x^2 - 5x + \dots$

1. c) $49a^6 - 28a^3b + \dots$

In a ceramic tile factory in Kilkenny, they design square tiles whose area is given by the trinomial $9x^2 + kx + 16$, where $k$ is an integer.

1. a) Deduce the two possible values that the coefficient $k$ can take so that the expression represents exactly the area of a square tile with side $(ax + b)$ or $(ax - b)$, with $a$ and $b$ being positive integers. In each case, write the associated squared binomial.
2. b) Taking the positive value of $k$ found, the designers are asked to cut out an interior decorative area by subtracting $25$ units from the total area, obtaining the expression $(3x + 4)^2 - 25$. Without expanding the squares, factor this last expression as the product of two first-degree binomials by applying the sum and difference identity, and simplify the numerical terms of each factor.

A solar panel manufacturer has discovered that the efficiency of its panels improves if a cell configuration is used that follows the difference of two squares identity (the product of a sum and a difference).

If the total area of a panel is given by the polynomial $A(x) = 16x^4 - 81y^2$, what are the algebraic expressions that represent the length and width of the panel?

Algebraic identities are also useful for performing quick mental calculations with large numbers. Use the difference of two squares identity (product of a sum and a difference) $(a + b) \times (a - b) = a^2 - b^2$ to calculate the following products cleverly:

1. a) $52 \times 48$ (Hint: consider $50 + 2$ and $50 - 2$)

1. b) $99 \times 101$

1. c) $35 \times 25$