Explain How To Factor When The Leading Coefficient Is 1

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Factoring When the Leading Coefficient is 1

I’ve always found factoring polynomials to be a tricky business, but I’ve learned that there are a few tricks that can make it a little easier. One of those tricks is knowing how to factor when the leading coefficient is 1. In this article, I’ll explain how to do just that.

Before we get started, let’s quickly review what a leading coefficient is. The leading coefficient is the coefficient of the term with the highest degree. For example, in the polynomial \(x^2 + 2x + 1\), the leading coefficient is 1.

Factoring a Polynomial with a Leading Coefficient of 1

When the leading coefficient of a polynomial is 1, there are two cases to consider:

1. The polynomial is a perfect square trinomial.
2. The polynomial is not a perfect square trinomial.

Case 1: The Polynomial is a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. For example, the polynomial \(x^2 + 2x + 1\) is a perfect square trinomial because it can be written as \((x + 1)^2\).

To factor a perfect square trinomial, we simply factor out the binomial. For example,

\(x^2 + 2x + 1 = (x + 1)^2\)

Case 2: The Polynomial is Not a Perfect Square Trinomial

If the polynomial is not a perfect square trinomial, we can still factor it by using the following steps:

1. Find two numbers that add up to the coefficient of the middle term and multiply to the product of the first and last coefficients.
2. Rewrite the middle term using these two numbers.
3. Factor by grouping.

For example, let’s factor the polynomial \(x^2 + 5x + 6\).

1. The coefficient of the middle term is 5, and the product of the first and last coefficients is 6.
2. The two numbers that add up to 5 and multiply to 6 are 2 and 3.
3. We can rewrite the middle term as 2x + 3x.
4. We can now factor by grouping:

\(x^2 + 5x + 6 = x^2 + 2x + 3x + 6\)

\(= (x^2 + 2x) + (3x + 6)\)

\(= x(x + 2) + 3(x + 2)\)

\(= (x + 3)(x + 2)\)

Tips and Expert Advice

Here are a few tips and advice from experts that can help you when factoring polynomials:

• Always check to see if the polynomial is a perfect square trinomial first.
• If the polynomial is not a perfect square trinomial, look for two numbers that add up to the coefficient of the middle term and multiply to the product of the first and last coefficients.
• Be patient and don’t give up if you don’t get it right the first time.

With a little practice, you’ll be able to factor any polynomial with a leading coefficient of 1.

FAQ

1. What is the leading coefficient?

The leading coefficient is the coefficient of the term with the highest degree.

2. How do I factor a polynomial with a leading coefficient of 1?

There are two cases to consider: the polynomial is a perfect square trinomial or it is not. If it is a perfect square trinomial, you can factor it out as the square of a binomial. If it is not a perfect square trinomial, you can factor it by using the following steps: find two numbers that add up to the coefficient of the middle term and multiply to the product of the first and last coefficients, rewrite the middle term using these two numbers, and factor by grouping.

3. What are some tips for factoring polynomials?

Always check to see if the polynomial is a perfect square trinomial first. If it is not, look for two numbers that add up to the coefficient of the middle term and multiply to the product of the first and last coefficients. Be patient and don’t give up if you don’t get it right the first time.

Conclusion

I hope this article has helped you learn how to factor polynomials with a leading coefficient of 1. If you have any questions, please feel free to leave a comment below.

Are you interested in learning more about factoring polynomials? If so, I encourage you to check out the following resources:

• Khan Academy: Factoring polynomials
• Math is Fun: Factoring polynomials
• Purplemath: Factoring polynomials

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