Polynomial Math: Multiply & Divide with Simple Animations

Welcome to understanding polynomials! Today we'll explore these fundamental expressions in algebra.A polynomial is a mathematical expression that follows specific rules. Let's break it down.Here's an example of a polynomial: two x squared plus three x plus one.Let's examine each term of our polynomial. The first term is two x squared.The second term is three x. Notice how the exponent of one is usually not written.The last term is simply one, which is a constant term with no variable.Let's organize what we've learned about polynomial structure in a clear table.The degree of a polynomial is determined by the highest exponent in any term. In this case, it's two.Here are some more examples of polynomials. Notice how they can have different numbers of terms and different degrees.When multiplying monomials, we follow two key rules: multiply the coefficients and add the exponents of like variables.Let's start with a simple example: three x squared times two x cubed.First, we separate the coefficients from the variables.Next, we multiply the coefficients: three times two equals six. For the variables, we add the exponents: two plus three equals five.This gives us our final answer: six x to the fifth power.Let's try a more complex example with two variables: negative two x cubed y, times three x y squared.Again, we separate coefficients and variables. Notice we have both x and y terms.Multiply the coefficients: negative two times three is negative six. For x, we add exponents three plus one. For y, we add exponents one plus two.Our final answer is negative six x to the fourth power y cubed.Let's solve one with a fraction: one-half x squared y cubed, times four x y.We multiply the coefficients: one-half times four equals two. Then add exponents for each variable separately.For x, we add two plus one. For y, we add three plus one.This gives us two x cubed y to the fourth power.For our final example, let's multiply terms with three variables: negative three x squared y squared z, times two x y z cubed.We multiply negative three and two to get negative six. Then we handle each variable separately.Add exponents for each variable: x is two plus one, y is two plus one, and z is one plus three.Our final answer is negative six x cubed y cubed z to the fourth power.The FOIL method helps us multiply two binomials step by step.Let's start with a simple example: x plus 2 times x plus 3.First, we multiply the first terms: x times x equals x squared.Next, multiply the outer terms: x times 3 equals 3x.For the inner terms: 2 times x equals 2x.Finally, multiply the last terms: 2 times 3 equals 6.Now we write out all terms: x squared plus 3x plus 2x plus 6.Notice that 3x and 2x are like terms. We can combine them: 3x plus 2x equals 5x.Let's try a more challenging example: 2x minus 1 times x plus 4.When we combine all terms, we get 2x squared plus 8x minus x minus 4.Combining like terms, 8x minus x equals 7x, giving us our final answer: 2x squared plus 7x minus 4.When dividing a polynomial by a monomial, we follow specific steps to simplify the expression.First, we distribute the division to each term in the numerator.For the coefficients, we divide the numbers. Six divided by three equals two, and three divided by three equals one.For the exponents, we subtract the powers. x cubed divided by x equals x squared, and x squared divided by x equals x to the first power.This gives us our simplified result: two x squared minus x.Let's look at a more complex example.Again, we start by distributing the division to each term.Let's break down each step of the simplification process.For the exponents, we subtract the power of x squared from each term.This gives us our final result: four x squared plus two x plus one.Let's review the general rules for dividing polynomials by monomials.Keep these rules in mind as we move on to more complex polynomial division.Let's work through polynomial long division step by step.First, we set up our problem like regular long division, with x squared plus three x plus two divided by x plus one.We start by dividing the first terms: x squared divided by x gives us x.Now multiply x times x plus one, giving us x squared plus x.Subtract x squared plus x from x squared plus three x plus two.Now we bring down the remaining terms and repeat the process with two x plus two.Multiply two times x plus one to get two x plus two.Subtract two x plus two from two x plus two, giving us zero remainder.Therefore, x squared plus three x plus two divided by x plus one equals x plus two with no remainder.Let's review the key points of polynomial long division.Thanks for learning polynomial long division with Spark.E!