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Factoring Polynomials Using Quadratic Form: Steps, Rules & Examples

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  • 0:01 Changing Form
  • 1:15 The Quadratic Form
  • 2:24 Changing to Quadratic Form
  • 4:45 Examples
  • 8:12 Lesson Summary
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Lesson Transcript
Instructor: David Liano
Factoring a polynomial of degree 4 or higher can be a difficult task. However, some polynomials of higher degree can be written in quadratic form, and the techniques used to factor quadratic functions can be utilized.

Changing Form

Factoring a polynomial, such as x4 - 29x2 + 100 might seem intimidating. In this lesson, you will learn how to change the form of certain polynomials of higher degree so that they are much easier to factor.

Let's first discuss changing form in the world of insects. Insects typically go through multiple life stages, and the appearance of some insects is very different from one life stage to the next. In other words, they change form. Consider the monarch butterfly: in its larval stage, it is in the form of a caterpillar. During this adolescent stage, the caterpillar is eating leaves like there is no tomorrow. At the right time, the monarch caterpillar prepares for its transformation to adulthood. It emerges from its shell to reveal a radical biological change. The caterpillar has changed form so that it can now enjoy the life and responsibilities of an adult butterfly.

In mathematics, we are usually not changing the form of living things. However, changing the forms of mathematical expressions might be effective in finding solutions to problems, like the one I mentioned at the beginning of the lesson.

The Quadratic Form

Let's look at the standard form of the quadratic function in one variable: y = ax2 + bx + c. A quadratic function in one variable has a degree of 2 because the variable of the leading term has an exponent of 2. The second term in x (bx) actually has an exponent of 1, but this exponent is generally not shown. The letters a, b, and c represent real numbers, except that a cannot equal zero.

Let's consider the following quadratic equation: x2 + 4x - 21 = 0. We can factor this equation as follows: (x + 7)(x - 3) = 0. We can now use the zero product property to solve the equation: x + 7 = 0, so x = -7. x - 3 = 0, so x = 3. However, the remainder of this lesson will focus on the task of factoring.

Changing to Quadratic Form

Now let's look at a polynomial expression that has a degree higher than two, as follows: x4 + x2 - 12. This expression does not fit our definition of a quadratic function in one variable because it has a degree of 4. However, we can rewrite it in quadratic form. Remember that the lead term in a quadratic expression has an exponent of 2 and the other term x has an exponent of 1. We can first rewrite the expression as follows:

(x2)2 + (x2)1 - 12

Let's do one more thing: let's substitute u for x2. In other words, u = x2. We now get the following: u2 + u - 12. There we have it: a quadratic expression! It is now much easier to factor. In factored form, it is (u + 4)(u - 3). Remember that u = x2. Now change u back to the term that it originally replaced: (x2 + 4)(x2 - 3). Our original expression is now factored:

x4 + x2 - 12 = (x2 + 4)(x2 - 3)

You might have noticed that if we square the second term in x, we get a term with an exponent of 4, just like the lead term. In other words, (x2)2 = x4. This will always be the case for polynomial expressions that can be written in quadratic form.

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