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A perfect square trinomial is a special type of polynomial that takes on the structure

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)

The exercise expression \(4x^2 - 28x + 49\) is indeed a perfect square trinomial. This means it can be perfectly expressed as the square of a binomial.
To recognize a perfect square trinomial, first check if the first and last terms are perfect squares. Here, both \(4x^2\) and \(49\) are perfect squares as we can write \(4x^2\) as \((2x)^2\) and \(49\) as \(7^2\). The middle term should equal the product of the square roots of the first and last terms, multiplied by 2. Specifically, \(2 imes 2x imes 7 = -28x\).
When all these conditions are met, you can express the trinomial in its factored form, which in this case is \((2x - 7)^2\). This is what helps to quickly and accurately solve the problem of factoring certain quadratics.

Factoring quadratics involves expressing a quadratic polynomial as a product of two binomials. Not all quadratics are the same, so various methods exist to factor them. These include:

• Factoring by Grouping: This method is often used when the quadratic equation looks complex but can be split into simpler parts.
• Using the Quadratic Formula: A formula used when factorization is not easily apparent.
• Recognizing a Perfect Square Trinomial: As discussed, if a quadratic matches the pattern of a perfect square trinomial, it can be easily expressed as \((ax + b)^2\) or \((ax - b)^2\).

The quadratic in our exercise was factored using the recognition of a perfect square trinomial. This saves time and makes the factoring precise, as opposed to dealing with longer methods like completing the square or using the quadratic formula for more complex quadratics.

An algebraic expression is a mathematical phrase that contains numbers, variables, and operation symbols. Quadratic expressions, like \(4x^2 - 28x + 49\), are one type of algebraic expression that specifically includes terms with an integer exponent of 2 as the highest degree.
Understanding algebraic expressions involves recognizing components, such as coefficients, constants, and variables. Each piece plays a distinct role. In our exercise:

• Coefficient: The numerical part in the term \(4x^2\) is 4, and in \(-28x\) it is -28.
• Variable: The letter \(x\) that represents a number is a variable.
• Constant: A number on its own, like 49 here.

Bringing all these parts together, algebraically manipulating them through methods like factoring quadratics allows us to simplify and solve expressions and equations effectively. This foundational skill is essential in algebra, laying the groundwork for more advanced math courses.