Solving Polynomials with Factor Theorem - Easy Guide

In this section, we'll learn about polynomials and the Factor Theorem.Let's begin by understanding what polynomials are. A polynomial is an expression with variables and coefficients, combined using addition, subtraction, and multiplication.Here are some examples of polynomials. P of x equals x cubed minus four x squared plus five x minus two. Q of x equals two x to the fourth plus three x squared minus seven.Let's examine the components of our polynomial P of x. Each term has a coefficient, a variable raised to a power, and a degree based on the exponent. The highest power of x determines the degree of the entire polynomial.Now, let's explore the Factor Theorem. The Factor Theorem states that x minus a is a factor of a polynomial P of x, if and only if P of a equals zero.This means if we can find a value a where P of a equals zero, then x minus a is a factor of our polynomial. In other words, a is a root or zero of the polynomial. This theorem provides us with a powerful method to factorize polynomials by finding their roots.Let's see an example using the polynomial P of x equals x cubed minus four x squared plus five x minus two. We'll evaluate P of 2.We substitute x equals 2 into the polynomial. P of 2 equals 2 cubed minus 4 times 2 squared plus 5 times 2 minus 2.2 cubed is 8, and 2 squared is 4.Computing each term: 8 minus 16 plus 10 minus 2.Adding these values, we get P of 2 equals 0.Since P of 2 equals 0, we can apply the Factor Theorem. It tells us that x minus 2 is a factor of our polynomial P of x.This means we can write P of x as the product of x minus 2 and another polynomial Q of x.Using polynomial division, we find that P of x equals x minus 2 multiplied by x squared minus 2x plus 1.To summarize what we've learned: A polynomial is an expression with variables and coefficients. The Factor Theorem tells us that x minus a is a factor of a polynomial if the polynomial evaluated at a equals zero. This powerful theorem helps us find factors by evaluating the polynomial at different values.In the next section, we'll learn how to find potential factors using the Rational Root Theorem.Having identified that x equals 1 is a root of our polynomial, we can use this to factor it.According to the Factor Theorem, if x equals 1 is a root, then x minus 1 is a factor of our polynomial.Now we need to divide our polynomial by this factor to find the remaining factors.Let's perform synthetic division to divide our polynomial by x minus 1.We bring down the first coefficient, multiply by 1, and add to the next coefficient. We continue this process for each term.The bottom row gives us our quotient coefficients, and the last number, zero, confirms that x equal 1 is indeed a root.So our quotient polynomial is x squared minus 5x plus 6.Now we need to factor this quadratic expression. Let's find two numbers that multiply to give 6 and add to give negative 5.We can split the middle term negative 5x into negative 2x and negative 3x since negative 2 times negative 3 equals 6, and negative 2 plus negative 3 equals negative 5.Now we can factor by grouping. We factor out x from the first two terms and negative 3 from the last two terms.We see that x minus 2 is a common factor, which gives us x minus 2 times x minus 3.Now we can write the complete factorization of our original polynomial.Each factor in our factorization corresponds to a root of the polynomial.The factor x minus 1 corresponds to the root x equals 1.The factor x minus 2 corresponds to the root x equals 2.And the factor x minus 3 corresponds to the root x equals 3.To summarize the factorization process:And that's how we factor a polynomial using its identified roots.In this section, we'll verify the factorization of a polynomial with practical examples.Let's start with the polynomial P of x equals x cubed minus six x squared plus eleven x minus six.We've factored this as the product of x minus one, x minus two, and x minus three.There are two main ways to verify our factorization is correct.Let's verify by multiplying the factors together.First, we multiply x minus one and x minus two using the FOIL method.This gives us x squared minus two x minus x plus two.Combining like terms, we get x squared minus three x plus two.Now, we multiply this result by x minus three.Distributing each term, we get x cubed minus three x squared, minus three x squared plus nine x, plus two x minus six.Grouping like terms, we have x cubed minus six x squared plus eleven x minus six.This result exactly matches our original polynomial, confirming our factorization is correct.Another way to verify is to evaluate the polynomial at each of its roots.Let's evaluate P of 1. We substitute x equals 1 into our polynomial.Simplifying, we get 1 minus 6 plus 11 minus 6, which equals zero.Similarly, when we evaluate P of 2, we get 8 minus 24 plus 22 minus 6, which also equals zero.And for P of 3, we get 27 minus 54 plus 33 minus 6, which equals zero as well.Let's summarize our verification methods.Here are some practical tips when working with polynomial factorization.Remember, verification is an essential step to ensure your polynomial factorization is correct. Always take time to check your work.