Exercises: 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) $(x^2 + 4) \cdot (x^2 - 4)$

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

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

Sophia and Ethan 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) $(x - 5) \cdot (x + 5) = x^2 - 25$

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

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

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

1. b) $99 \cdot 101$

1. c) $35 \cdot 25$

Copy and complete in your notebook 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) $16x^2 - \_\_\_\_ = (\_\_\_\_ + 3y) \cdot (\_\_\_\_ - 3y)$

Simplify the following algebraic expression fully by first expanding the algebraic identities and then combining like terms:

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

A city council wants to design a new square plaza. In the center there will be a landscaped area, also square, surrounded by a tiled path of constant width.

If the total side of the plaza measures $(3x + 2)$ feet and the side of the central garden measures $(3x - 2)$ feet, calculate the simplified polynomial that represents the area intended solely for the tiled path.

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

1. a) Express the area of a face as a squared binomial if the edge were to increase 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 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 value of the perimeter of the hexagon for $x = 4\text{ cm}$ (simplify the root).

Match in your notebook each algebraic identity 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 equalities are correct:

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

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

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

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

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 that represents the area.

A very common mistake when starting with algebraic identities is to think 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 the same or different.

Expand the following expressions using algebraic identities and combine 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 (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 factor 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 of $x$ pixels. The client requests two changes: first, to increase the side by $4$ pixels to add a decorative border; second, to 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 those 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 $2x-3$.

Calculate 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 their panels improves if a cell configuration is used that follows the identity of 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 this panel?