Problem 55 - Elementary and Intermediate Algebra

Question

Factor. $$ -49 d^{4}+16 c^{2} $$

Step-by-Step Solution

Answer

The factorized form is \[(4c - 7d^2)(4c + 7d^2)\].

Step 1: Recognize the Form

The expression \[-49d^4 + 16c^2\]is a difference of squares because it can be rewritten as \[-(7d^2)^2 + (4c)^2\].Recall that a difference of squares can be factored as:\[(a^2 - b^2) = (a - b)(a + b)\].

Step 2: Identify Components

Notice:- \((7d^2)^2\) means `a` in our formula is \(7d^2\).- \((4c)^2\) means `b` in our formula is \(4c\).We can now apply the difference of squares.

Step 3: Apply the Difference of Squares Formula

Use the formula \((a^2 - b^2) = (a - b)(a + b)\) to rewrite the expression:\[-49d^4 + 16c^2 = -((7d^2)^2 - (4c)^2)\].This can be factored further as:\[-((7d^2) - (4c))((7d^2) + (4c))\].

Step 4: Simplify the Factorization

Distribute the negative sign outside to get the final factorization:\[(4c - 7d^2)(4c + 7d^2)\].

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It involves rewriting a polynomial as a product of simpler expressions called "factors." For example, the polynomial \(-49d^4 + 16c^2\) can be rewritten into simpler terms. The goal of factoring is often to simplify an expression or to solve equations more easily.
A common method for factoring is identifying patterns. Some patterns, like the difference of squares, are especially useful. Recognizing these patterns allows you to transform the expression into a product of terms, which might be easier to work with.

• Look for a difference of squares, which typically has the form \(a^2 - b^2\).
• Once identified, use the identity \((a-b)(a+b)\) to factor them.
• Practicing this technique will make factoring more intuitive over time.

Factoring not only simplifies polynomials but also aids in solving equations, simplifying rational expressions, and more. It's a fundamental tool in a mathematician's toolkit.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra. Expressions like \(-49d^4 + 16c^2\) represent more than just numbers; they encapsulate relationships between variables. Understanding these is crucial for advancing in mathematics.
In algebra, expressions are often manipulated to solve problems:

• Adding, subtracting, and simplifying expressions are common tasks.
• Expressions can be categorized, like linear, quadratic, or polynomial expressions.
• They can be rewritten in various forms, such as factored form, by using techniques like factoring.

Grasping how to work with algebraic expressions solidifies your ability to tackle complex mathematical problems and equations.

Quadratic Expressions

Quadratic expressions are polynomials of the second degree. They are of the form \(ax^2 + bx + c\). These expressions are fundamental in algebra, as they appear frequently in mathematical problems.
The essence of quadratic expressions lies in their parabolic graphs and the symmetry they exhibit.

• Quadratic expressions have two solutions, found through factoring, completing the square, or using the quadratic formula.
• Factoring quadratics involves finding two binomials whose product is the quadratic expression.
• The difference of squares is a special case where the quadratic expression can be factored quickly if recognized.

Mastering quadratic expressions involves understanding both their algebraic manipulation and their graphical representation, enabling deeper insights into how they behave and how they can be solved in various contexts.