Expand each expression using the Binomial Theorem. (a x-b...

Key Concepts

Handling Negative Terms in Binomials

When a binomial expression includes a negative term, such as in (ax - by)^n, the negative sign is raised to a power along with its accompanying variable term. This results in alternating sign patterns in the expanded form, depending on whether the exponent associated with the negative term is even or odd. Understanding this helps in correctly determining the sign of each term during the expansion process.

Polynomial Expansion

Polynomial expansion involves expressing an expression raised to a power as a sum of terms, each of which is a product of powers of the individual variables. This process makes use of formulas like the Binomial Theorem to systematically derive each term, providing a clear and organized method to handle higher powers of polynomial expressions.

Binomial Theorem

The Binomial Theorem is a formula used for expanding expressions that are raised to a power, specifically expressions of the form (x + y)^n. It provides a systematic way to expand the expression by summing terms that involve binomial coefficients and various powers of the individual terms, simplifying the process of expansion for any non-negative integer exponent.

Binomial Coefficients

Binomial coefficients are numerical factors that appear in the expansion of a binomial raised to a power. They are given by the formula 'n choose k', representing the number of ways to choose k elements from a set of n elements. These coefficients not only determine the weights of each term in the expanded expression but also connect to important combinatorial principles.

Transcript

00:01 We can expand the binomial a x minus b y to the fourth using the binomial theorem.

00:07 The first step is to rewrite this binomial.

00:13 We can rewrite this binomial as a x plus negative b y all to the fourth.

00:25 When the equations in this form, we can see that the x over here in the binomial theorem corresponds to a x in our equation.

00:34 A over here corresponds to negative by and n over here equals 4.

00:47 So using the binomial theorem we can see that the first term is n choose j.

00:53 For the first term, j equals 0.

00:57 So we get n which is 4, choose j which is 0, times x, which is in this case, a x for for our equation to the n minus j in this term n equals 4 j equals 0 so be a x to the 4th times a to the j now remember j equals 0 and anything to the 0 power just equals 1 for the next term j equals 1 so we get 4 choose 1 times a x to the 4 minus 1 so to the 3rd times negative b y to the 4 for the next term, j equals 2.

01:48 So we get 4 choose 2 times a x to the 4 minus 2.

01:55 So a x squared times negative b y squared.

02:02 The next term, j equals 3.

02:05 So we get 4 choose 3 times a x to the 1st times negative b y to the 3.

02:21 Now for the last term, j equals 4.

02:24 So we get 4 choose 4 times a x to the 4 minus 4.

02:31 So a x to the 0, which equals 1, times negative b y to the 4th...

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