How to Factor Polynomial with Leading Coefficient
Polynomial factorization is a fundamental skill in algebra that involves expressing a polynomial as a product of simpler polynomials. One of the most common types of polynomials is the one with a leading coefficient, which is the coefficient of the highest degree term. Factoring such polynomials can be challenging, especially for those who are new to the subject. In this article, we will discuss various methods to factor polynomials with leading coefficients, helping you understand the process and become more proficient in polynomial factorization.
1. Factor by grouping
One of the simplest methods to factor a polynomial with a leading coefficient is by grouping. This method involves grouping terms with common factors and then factoring out those common factors. Here’s a step-by-step guide:
1. Arrange the polynomial in descending order of degrees.
2. Group the terms with common factors.
3. Factor out the common factors from each group.
4. Multiply the factored expressions from each group to obtain the final factored form.
For example, consider the polynomial \(2x^3 – 3x^2 + 5x – 6\). To factor this polynomial by grouping, we can group the terms as follows:
\((2x^3 – 3x^2) + (5x – 6)\)
Now, factor out the common factors from each group:
\(x^2(2x – 3) + 1(5x – 6)\)
Next, we can factor out the greatest common factor (GCF) from the entire expression:
\((x^2 + 1)(2x – 3)\)
So, the factored form of the polynomial is \((x^2 + 1)(2x – 3)\).
2. Factor by finding a zero
Another method to factor a polynomial with a leading coefficient is by finding a zero. This method involves finding a value for the variable that makes the polynomial equal to zero. Once we find a zero, we can factor out the corresponding linear factor and continue factoring the remaining polynomial.
Here’s a step-by-step guide:
1. Arrange the polynomial in descending order of degrees.
2. Find a zero for the polynomial by using synthetic division, the Rational Root Theorem, or by trial and error.
3. Factor out the linear factor corresponding to the zero.
4. Continue factoring the remaining polynomial until it cannot be factored further.
For example, consider the polynomial \(x^3 – 6x^2 + 11x – 6\). We can find a zero for this polynomial by using synthetic division or the Rational Root Theorem. Let’s assume we find that \(x = 1\) is a zero. Now, we can factor out the linear factor \((x – 1)\):
\((x – 1)(x^2 – 5x + 6)\)
Next, we need to factor the remaining quadratic polynomial. We can factor it by grouping or by finding a zero for the quadratic. In this case, let’s factor it by grouping:
\((x – 1)(x – 2)(x – 3)\)
So, the factored form of the polynomial is \((x – 1)(x – 2)(x – 3)\).
3. Factor by completing the square
Completing the square is another method to factor a polynomial with a leading coefficient. This method involves rewriting the polynomial in the form of a perfect square trinomial. Here’s a step-by-step guide:
1. Arrange the polynomial in descending order of degrees.
2. Move the constant term to the right side of the equation.
3. Divide the coefficient of the \(x^2\) term by 2 and square it.
4. Add the squared value to both sides of the equation.
5. Rewrite the left side of the equation as a perfect square trinomial.
6. Factor the perfect square trinomial.
7. Solve for the variable.
For example, consider the polynomial \(x^2 – 4x + 3\). To factor this polynomial by completing the square, we can follow these steps:
1. Move the constant term to the right side of the equation:
\(x^2 – 4x = -3\)
2. Divide the coefficient of the \(x\) term by 2 and square it:
\(\left(\frac{-4}{2}\right)^2 = 4\)
3. Add the squared value to both sides of the equation:
\(x^2 – 4x + 4 = 1\)
4. Rewrite the left side of the equation as a perfect square trinomial:
\((x – 2)^2 = 1\)
5. Factor the perfect square trinomial:
\((x – 2)^2 – 1 = 0\)
6. Solve for the variable:
\(x – 2 = \pm1\)
\(x = 2 \pm 1\)
So, the factored form of the polynomial is \((x – 3)(x – 1)\).
In conclusion, factoring polynomials with leading coefficients can be achieved using various methods such as grouping, finding a zero, and completing the square. By understanding these methods and practicing them, you can become more proficient in polynomial factorization and apply your skills to solve more complex algebraic problems.
