Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \((x+3)^{5}-4(x+3)^{4}\)

Step-by-Step Solution

Answer

The expanded and simplified form of the expression \((x+3)^{5}-4(x+3)^{4}\) is therefore \(x^{5}+11x^{4}+42x^{3}-162x^{2}-891x-729\).

Step 1: Expanding using the Binomial Theorem

Firstly expand both \( (x+3)^{5} \) and \( 4(x+3)^{4} \) using the formula from the Binomial Theorem. \((x+3)^{5}={5 \choose 0}x^{5}3^{0}+{5 \choose 1}x^{4}3^{1}+{5 \choose 2}x^{3}3^{2}+{5 \choose 3}x^{2}3^{3}+{5 \choose 4}x^{1}3^{4}+{5 \choose 5}x^{0}3^{5}\). Calculate each term and simplify. \((x+3)^{5}=x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243\). \n Secondly, apply Binomial theorem to \( 4(x+3)^{4} \). Notice the factor 4 in front of the second term, which gives us \( 4(x+3)^{4}\). Expanding the term: \((x+3)^{4}={4 \choose 0}x^{4}3^{0}+{4 \choose 1}x^{3}3^{1}+{4 \choose 2}x^{2}3^{2}+{4 \choose 3}x^{1}3^{3}+{4 \choose 4}x^{0}3^{4}\) and simplify to \( (x+3)^{4}=x^{4}+12x^{3}+108x^{2}+324x+243\). Multiply the result by 4, which gives: \(4(x+3)^{4}=4x^{4}+48x^{3}+432x^{2}+1296x+972\)

Step 2: Subtraction

Since we want to compute \((x+3)^{5}-4(x+3)^{4}\), we subtract the expanded form of the second expression from that of the first one. This gives: \(x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243 - (4x^{4}+48x^{3}+432x^{2}+1296x+972)\). Simplify this to get: \(x^{5}+11x^{4}+42x^{3}-162x^{2}-891x-729\)

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra that allows us to expand expressions that are raised to a power, such as \((x+3)^5\). The Binomial Theorem provides a formula to expand binomials of the form \((a+b)^n\). According to the theorem:

• Each term in the expansion is made up of a binomial coefficient \({n \choose k}\),
• multiplied by \(a\) raised to a certain power,
• and \(b\) raised to a complementary power.

For instance, in the expression \((x+3)^5\), \(x\) is \(a\) and \(3\) is \(b\). By calculating the combination of binomial coefficients and powers, you generate each term in the expanded polynomial. This process results in a polynomial consisting of several terms depending upon the original binomial and the power to which it is raised. For example, expanding \((x+3)^5\) yields six terms, starting from \(x^5\) and ending with a constant.

Combining Like Terms

Once you have expanded a polynomial or multiple polynomials, the next step is combining like terms. Like terms are those that have the same variable raised to the same power. They can be added or subtracted from each other to simplify the expression. For example:

• Suppose you have terms like \(15x^4\) and \(4x^4\).
• These are like terms because they both have \(x^4\), thus they can be combined into \(11x^4\).

In subtraction operations, ensure each type of term is properly combined, as it will change the coefficients. For example, if poly_expansion is \(x^5 + 15x^4 + 90x^3\) and you subtract \(4x^4 + 48x^3\), the resulting terms would be \(x^5 +(15x^4-4x^4) + (90x^3-48x^3)\), leading to \(x^5 + 11x^4 + 42x^3\). The concept requires careful attention to ensure each like term is calculated accurately.

Simplification in Algebra

Simplifying an algebraic expression involves reducing it to its simplest form while maintaining equivalency. This process often follows after combining like terms. Simplification helps in making calculations easier and expressions clearer. Here are some aspects of algebraic simplification:

• Reduce individual terms: After combining terms, check if each term can be simplified further. This may sometimes require factoring.
• Check for common factors: If the entire expression can be divided by a common factor, do so to simplify it.
• Re-arrange: Place terms to reduce potential errors and improve readability.

For example, in the expanded result of \((x+3)^5 - 4(x+3)^4\), after combining and rearranging terms, you are left with \(x^5 + 11x^4 + 42x^3 - 162x^2 - 891x - 729\). While this may be as simplified as it could be in this context, ensuring minimum complexity is key to ease of interpretation and further algebraic manipulation.