Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(1-4 x+4 x^{2}\)

Step-by-Step Solution

Answer

The factored form of \(1-4 x+4 x^{2}\) is \((2x-1)^2\)

Step 1: Write the quadratic function in a standard form

Make sure the function is written standard form, which is \(ax^2+bx+c\). The given function is \(4x^2-4x+1\).

Step 2: Compare with the formula

Compare this function with the complete square formula which is \((px+q)^2\). On comparison, we see that \(p^2=4, 2pq=-4\) and \(q^2=1\). From these equations, we conclude that \(p=2\) and \(q=1\) as the equations fit best in these scenarios.

Step 3: Write factored form

Now that we know values of \(p\) and \(q\), it is easy to write the given function in a factored form. So the function \(4x^2-4x+1\) becomes \((2x-1)^2\).

Key Concepts

Completing the Square

Completing the square is a nifty mathematical technique used to convert a quadratic expression into a perfect square trinomial. It’s particularly helpful when you're trying to factor quadratic expressions or solve quadratic equations. Here’s a simple breakdown of how this process works:

• Start with a quadratic expression in the standard form of \(ax^2 + bx + c\).
• Rearrange the expression, focusing only on the terms involving \(x\). Don't worry about the constant yet.
• Create a perfect square trinomial by finding a value that makes the expression a square of a binomial.

For example, for the quadratic expression \(4x^2 - 4x + 1\), you can tweak it into \((2x-1)^2\). This is done by matching the expression to \((px+q)^2\), where \(p\) and \(q\) are chosen so that \(p^2=4\), \(2pq=-4\), and \(q^2=1\). This lets you transform and simplify a quadratic equation or expression more easily.

Quadratic Functions

Quadratic functions are polynomial functions of degree two. This means the highest power of the variable (often \(x\)) is squared. Quadratic functions typically look like this:\[f(x) = ax^2 + bx + c\] where:

• \(a\), \(b\), and \(c\) are constants.
• \(a\) is not equal to zero.
• The graph of a quadratic function is a parabola.

The direction of the parabola (whether it opens upwards or downwards) depends on the sign of \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

Quadratic functions are crucial in many areas of mathematics because they describe various natural phenomena and can model physical situations. By completing the square, factoring, or using the quadratic formula, you can solve for the roots of the quadratic equation, which are the values of \(x\) where \(f(x) = 0\). Understanding and recognizing the standard formula of quadratics is key to manipulating and solving these expressions.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into its simplest components, called factors. This process plays a crucial role, especially in solving equations and understanding their properties. Here's a breakdown of the factoring process:

• Identify the type of polynomial. For quadratics, this is often \(ax^2 + bx + c\) form.
• Determine if the quadratic can be expressed as a product of simpler polynomial factors.
• Use methods like factoring by grouping, using special products, or completing the square to simplify the expression.

In the example \(1-4x+4x^2\), rewriting it as \(4x^2 - 4x + 1\) allows us to express it as \((2x-1)^2\), showing that it factors completely into a binomial squared. Factoring helps in graphing polynomial functions, finding roots, and simplifying expressions for further mathematical operations. Mastering factoring techniques is invaluable as it fosters a deeper understanding of the relationships and structures within algebraic expressions.