Problem 66 - Algebra 1: Concepts and Skills

Question

Which of the following is a correct factorization \( \text { of }-12 x^{2}+147 ?\) A. \(-3(2 x+7)^{2}\) B. \(3(2 x-7)(2 x+7)\) C. \(-2(2 x-7)(2 x+7)\) D. \(-3(2 x-7)(2 x+7)\)

Step-by-Step Solution

Answer

The correct factorization of the expression \(-12x^2+147\) is \(-3(2x-7) (2x+7)\), which corresponds to option D.

Step 1: Factorising the Negative term

Start by moving the negative operator to outside the parentheses. This transforms \(-12x^2+147\) into \(-1(12x^2-147)\).

Step 2: Applying the difference of squares

Next apply the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\). Here \(a= \sqrt{12}x\) and \(b= \sqrt{147}\). So \(-1(12x^2-147)\) becomes \(-1( (\sqrt{12}x)^2 - (\sqrt{147})^2 ) = -1( (\sqrt{12}x - \sqrt{147}) (\sqrt{12}x + \sqrt{147}) )\).

Step 3: Simplify the Solution

Simplify the expression further as \(-1( (2\sqrt{3}x - 7\sqrt{3}) (2\sqrt{3}x + 7\sqrt{3}) )\) which becomes \(-1(2x-7\sqrt{3})(2x+7\sqrt{3})\). The-\(\sqrt{3}\) inside the brackets can be simplified further, giving us \(-1(2x-7) (2x+7)\). This will transform into \(-1*3* (2x-7) (2x+7) = -3(2x-7) (2x+7)\).

Key Concepts

difference of squares

The difference of squares is a fundamental concept in algebra that simplifies the factorization of certain quadratic expressions. It follows the formula: \[a^2 - b^2 = (a - b)(a + b)\]. This is powerful because it allows us to factor expressions that appear complex at first glance.
For the expression \(12x^2 - 147\), identify that it fits the difference of squares format where \(a = \sqrt{12}x\) and \(b = \sqrt{147}\). Applying the formula, the expression becomes:

• \(( sqrt{12}x - sqrt{147}) ( sqrt{12}x + sqrt{147})\)

Simplifying further involves recognizing common factors, making it easier to express in a more manageable form. This approach not only helps in factorizing but also in understanding relationships within expressions.

polynomials

Polynomials are expressions consisting of variables (like \(x\)), coefficients, and exponents that are combined using addition, subtraction, and multiplication. A polynomial is typically expressed as:
\[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]
Here, each \(a_n\) represents a coefficient, and \(x^n\) indicates the degree. In this case, \(-12x^2 + 147\) is a second-degree polynomial due to the \(x^2\) term.

• The highest degree term indicates the polynomial's order, which influences its graph's shape and turning points.
• Understanding polynomials also aids in various operations such as addition, subtraction, multiplication, division, and factorization.

Recognizing and manipulating these expressions is crucial for tackling algebraic equations and exploring mathematical relationships.

algebraic expressions

Algebraic expressions are statements comprised of numbers, variables, and arithmetic operations such as addition (+), subtraction (-), multiplication (*), and division (÷). They are fundamental in outlining mathematical relationships and allowing calculations beyond simple arithmetic.
For example, in the expression \(-12x^2 + 147\), we have two key elements:

• Terms: \(-12x^2\) (a term with variable and exponent) and \(+147\) (a constant term)
• Operators: Indicate the mathematical operations linked to these terms.

Algebraic expressions also form the basis for polynomials. Understanding and simplifying these expressions are pivotal for more complex topics like equations and inequalities. They serve as the backbone of solving real-world problems using math.