Problem 20 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (y-4)^{4} $$

Step-by-Step Solution

Answer

(y-4)^4 = y^4 - 16y^3 + 96y^2 - 256y + 256

Step 1: Expressing the problem

We have the binomial expression (y−4)^4 and we need to use the binomial theorem to expand the expression. The binomial theorem for a binomial expression (a+b)^n is given by: (a+b)^n = Σ (n choose k) * a^(n-k) * b^k. Here, a=y, b=-4, and n=4.

Step 2: Calculating the binomial coefficients

We need to calculate the binomial coefficients for k from 0 to 4. (n choose 0) = 4 choose 0 = 4!/[(4-0)! * 0!] = 1,(n choose 1) = 4 choose 1 = 4!/[(4-1)! * 1!] = 4,(n choose 2) = 4 choose 2 = 4!/[(4-2)! * 2!] = 6,(n choose 3) = 4 choose 3 = 4!/[(4-3)! * 3!] = 4,(n choose 4) = 4 choose 4 = 4!/[(4-4)! * 4!] = 1.

Step 3: Substituting the binomial coefficients into the binomial theorem

We will substitute the calculated binomial coefficients, values of a, b and n into the binomial theorem equation and expand the equation: (y-4)^4 = (n choose 0) * y^(4-0) * (-4)^0 + (n choose 1) * y^(4-1) * (-4)^1 + (n choose 2) * y^(4-2) * (-4)^2 + (n choose 3) * y^(4-3) * (-4)^3 + (n choose 4) * y^(4-4) * (-4)^4.

Step 4: Simplifying the equation

Simplify to obtain the final expanded form of the equation: (y-4)^4 = 1*y^4*(-4)^0 - 4*y^3*(-4)^1 + 6*y^2*(-4)^2 - 4*y^1*(-4)^3 + 1*y^0*(-4)^4 (y-4)^4 = y^4 - 4*4*y^3 + 6*16*y^2 - 4*64*y + 256 (y-4)^4 = y^4 - 16y^3 + 96y^2 - 256y + 256

Key Concepts

Binomial Expansion

The binomial expansion is a powerful mathematical technique used to expand expressions that are raised to a power. Essentially, it allows us to express a binomial expression like \((a + b)^n\) as a sum of terms that combine parts of the binomial in different powers.

The general formula for the binomial expansion is given by:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{(n-k)} b^k\)

In our exercise, the expression \((y-4)^4\) is expanded using this formula where \(a = y\), \(b = -4\), and \(n=4\).

We substitute these values to form a series of terms. Each term uses a binomial coefficient and the variables \(a\) and \(b\) raised to specific powers, which we will find through calculation. This method simplifies complex multiplications, making it manageable step by step.

Binomial Coefficients

When expanding a binomial expression using the binomial theorem, we encounter binomial coefficients. These are constants that appear in the binomial expansion formula. They are denoted as \(\binom{n}{k}\), which is read as "n choose k".

Their role is crucial because they determine the weight each term contributes to the overall expanded expression. Mathematically, binomial coefficients can be calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{(n-k)!k!}\)

In our specific exercise, we calculate the binomial coefficients for \(n = 4\):

• \(\binom{4}{0} = 1, \binom{4}{1} = 4, \binom{4}{2} = 6, \binom{4}{3} = 4, \binom{4}{4} = 1\).

These coefficients are then used in conjunction with the binomial expansion formula to determine each term of the expanded polynomial.

Polynomial Simplification

After applying the binomial theorem, we end up with a series of terms that need to be simplified. This step involves combining like terms and performing any necessary arithmetic operations to present the polynomial in its simplest form.

In the example with \((y-4)^4\), after using the calculated coefficients and substituting \(y\) and \(-4\) raised to their respective powers, we end up with the terms. Each term needs to be simplified by:

• Evaluating powers of numbers like \((-4)^k\).
• Multiplying coefficients with the outcomes of these powers.
• Arranging them as a standard polynomial expression, in decreasing order of power.

The expansion process ultimately simplifies to this form: \(y^4 - 16y^3 + 96y^2 - 256y + 256\). This final expression is a straightforward polynomial, much simpler to work with compared to the starting binomial.