Exercises: Standard algebraic identities

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

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.

Lucía and Diego are doing their math homework and have made some common mistakes. Identify which of the following equations is incorrect and correct it in your notebook, 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) $(2a - 3)^2 = 4a^2 - 12a + 9$

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) $35 \times 25$

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

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

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

A local municipality wants to design a new square town square. In the center 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 $(3x + 2)$ meters and the side length of the central garden is $(3x - 2)$ meters, calculate the simplified polynomial that represents the area intended solely for the paved path.

In the cube in the figure, with edge length $x$, the face diagonal of one of its faces and the main body diagonal have been drawn. In addition, a regular hexagon is formed by cutting the cube with a plane $P$ that passes through the midpoints of its edges.

1. a) Express as a squared binomial the area of a face if the edge length were increased by $2\text{ cm}$.
2. b) If the area of the hexagon can be expressed as $\frac{3\sqrt{3}}{4} x^2$, prove using algebraic identities that if the edge length were $(x+y)$, the new area of the hexagon would be $\frac{3\sqrt{3}}{4}(x^2 + 2xy + y^2)$.
3. c) Calculate the exact perimetre of the hexagon for $x = 4\text{ cm}$ (simplify the square root).

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

• A. $(x + 2)^2$

• B. $(x - 2)^2$

• C. $(x + 2)(x - 2)$

• $x^2 - 4$

• $x^2 + 4x + 4$

• $x^2 - 4x + 4$

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

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

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.

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.

Expand the following expressions using algebraic identities and collect like terms wherever possible:

1. a) $\left( \frac{3}{2}x^2 - \frac{4}{5} \right)^2$

1. b) $(2x^3 + 5y)^2 - (2x^3 - 5y)^2$

1. c) $(x^2 - 3)(x^2 + 3) - (x^2 - 1)^2$

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

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

An app designer is creating a square icon for a social network. The original design has a side length of $x$ pixels. The client asks for two changes: first, increase the side length by $4$ pixels to add a decorative border; second, create a reduced version where each side measures $4$ pixels less than the original.

Find the polynomial that represents the difference in area between the enlarged icon and the reduced icon in terms of $x$.

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 $x+1$).

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 $2x-3$.

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.

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?