Question

In Exercises 43–48, use Pascal’s Triangle to expand the binomial. \((y z+1)^5\)

Short answer

The expansion of \((y z + 1)^5\) using Pascal's Triangle is \( y^5 z^5 + 5 y^4 z^4 + 10 y^3 z^3 + 10 y^2 z^2 + 5 y z + 1 \)

Step-by-step solution

Identify the row from Pascal's Triangle

The binomial is raised to the power 5. So, you need to find the coefficients corresponding to the 5th row in the Pascal's Triangle. Start counting rows from 0. So the 5th row is \'1 5 10 10 5 1\'

Expanding the binomial using the coefficients

The expanded form of the binomial is given by \( (y z + 1)^5 = 1*(y z)^5 + 5*(y z)^4*1 + 10*(y z)^3*1^2 + 10*(y z)^2*1^3 + 5*(y z)^1*1^4 + 1*(y z)^0*1^5 \)

Simplify the Expression

Simplifying the above expression, you get the expanded form of the given binomial as \( y^5 z^5 + 5 y^4 z^4 + 10 y^3 z^3 + 10 y^2 z^2 + 5 y z + 1 \)

Key concepts

Pascal's Triangle

Pascal's Triangle is a magical and straightforward tool that makes the task of binomial expansion relatively simple. To understand it, imagine a triangle where each row represents the coefficients of the expanded form of a binomial expression raised to a certain power.The triangle begins with a '1' at the top, known as Row 0. Each subsequent row is constructed by adding the two numbers above it to form the next number in the new row. If there's only one number above, you add '1' to it. This way, the Triangle builds up, with each row providing the coefficients for the binomial expansion.For example, consider the 5th row for the binomial expansion of \((yz + 1)^5\). Here the row you look at is '1 5 10 10 5 1', correlating to the powers of each term in the binomial expression when expanded. By following Pascal's Triangle, one can avoid cumbersome calculations and quickly identify the coefficients needed for the expansion.

Polynomial Expressions

Polynomial expressions are algebraic terms composed of variables and coefficients, brought together by addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial expression is \(3x^2 + 2x + 1\). The power of the variable, in this case '2' for the term \(3x^2\), tells us the degree of the polynomial.Once we start expanding binomials like \((yz + 1)^5\), the result is a polynomial expression. This expansion process involves applying the distributive property over and over again until every term is simplified. Using tools like Pascal's Triangle streamlines this process by providing the coefficients directly, thus turning a potentially complex multiplication and distribution task into a simpler and more manageable one.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that makes equations and expressions easier to work with. It involves reducing expressions to their simplest form while maintaining their original value. This can be achieved by combining like terms, which are terms that have the same variable raised to the same power, and performing operations according to the order of operations.In the context of binomial expansion, simplifying expressions is the final step. After using Pascal's Triangle to determine the coefficients and writing out each term of the expanded binomial, you combine like terms, if any, and perform any necessary multiplications. For the given exercise \((yz + 1)^5\), the simplification step involves raising each term in the binomial to the relevant power and then multiplying it by the associated coefficient from Pascal's Triangle, leading us to the final simplified polynomial.