Problem 37 - College Algebra

Question

For the following exercises, multiply the binomials. $$(11 q-10)(11 q+10)$$

Step-by-Step Solution

Answer

The expanded expression is \(121q^2 - 100\).

Step 1: Identify the Structure

The expression given, \((11q -10)(11q + 10)\), is a product of two binomials that follow the pattern of \((a-b)(a+b)\). This can be expanded using the formula \((a-b)(a+b) = a^2-b^2\).

Step 2: Identify Values for a and b

Recognize that in the given binomials, \(a = 11q\) and \(b = 10\). These values will be used to apply the formula for the difference of squares.

Step 3: Apply the Difference of Squares Formula

Substitute the known values into the difference of squares formula: \(a^2 - b^2 = (11q)^2 - 10^2\).

Step 4: Calculate Each Term

Calculate \((11q)^2 = 121q^2\) and \(10^2 = 100\). Then substitute these values back into the expression: \(121q^2 - 100\).

Step 5: Simplify the Expression

The expression \(121q^2 - 100\) is already simplified, as there are no like terms to combine further.

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity that simplifies the multiplication of binomials that share a certain structure. If you have two binomials of the form

• \((a - b)\) and \((a + b)\)

, they can be quickly expanded to a single expression. The identity formula for the difference of squares is:\[(a - b)(a + b) = a^2 - b^2\]This means you only need to square the first part \(a\), then subtract the square of the second part \(b\). This simplification comes in handy as it requires fewer steps than multiplying each term of the binomials individually.
If you spot this pattern, using the identity saves both time and effort in solving algebra problems.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving variables raised to whole number exponents, combined by addition, subtraction, and multiplication.

• They can include terms like \(x^2\), \(3x\), or constants like 5.
• Polynomials can be represented with multiple terms, each called a monomial if it houses only one term.

In our problem, the result after applying the difference of squares to the binomials is the polynomial \(121q^2 - 100\).
This is a simplification of the original expression, showing the power of polynomial manipulation in algebra.
By observing and identifying the structure within polynomials, one can organize and simplify expressions more efficiently.

Algebraic Identities

Algebraic identities are pre-established rules in mathematics that express equality and allow simplification of expressions. A notable subset of algebraic identities includes patterns like

• the "difference of squares"
• "perfect square trinomials"
• and "sum and difference cubes"

. Knowing these identities is crucial to efficiently solving algebra problems. In the given exercise, the difference of squares identity \[(a - b)(a + b) = a^2 - b^2\]plays a pivotal role.
This identity is directly applied rather than expanding the binomials fully, showcasing how algebraic identities simplify the computational process and make complicated problems less daunting to tackle.
Overall, understanding and mastering these identities equips students with powerful tools to handle various algebraic challenges.