Problem 61 - Algebra Form and Function

Question

Given that \(\left(3 x^{3}+a\right)\left(2 x^{b}+3\right)=6 x^{7}+c x^{4}+d x^{3}+3\), find possible values for the constants \(a, b, c,\) and \(d\).

Step-by-Step Solution

Answer

Question: Determine the values of the constants a, b, c, and d in the given multiplication of expressions: (3x^3 + a)(2x^b + 3) = 6x^7 + cx^4 + dx^3 + 3. Answer: The values of the constants are a = 1, b = 4, c = 2, and d = -9.

Step 1: Expansion of the given expressions

Expand the given expressions: \(\left(3x^3 + a\right)\left(2x^b + 3\right)\). This can be done by applying the distributive property (also known as FOIL), by multiplying each term in the first expression with each term in the second expression: \((3x^3 + a)(2x^b + 3) = (3x^3)(2x^b) + (3x^3)(3) + a(2x^b) + a(3)\)

Step 2: Simplification of the expanded expression

Simplify the expanded expression by combining like terms: \((3x^3)(2x^b) = 6x^{3+b}\), \((3x^3)(3) = 9x^3\), \(a(2x^b) = 2ax^b\), and \(a(3) = 3a\). So, the simplified expression is: \(6x^{3+b} + 9x^3 + 2ax^b + 3a\).

Step 3: Comparing the two polynomials

Compare the simplified expression with the given polynomial: \(6x^{7} + cx^4 + dx^3 + 3 = 6x^{3+b} + 9x^3 + 2ax^b + 3a\).

Step 4: Matching coefficients of corresponding terms

Match the coefficients of the corresponding terms: 1. Coefficient of \(x^{3+b}\): \(6 = 6\), which implies that \(3+b = 7\) and solving for \(b\), we get \(b = 4\). 2. Coefficient of \(x^3\): \(d + 9 = 0\), we will solve for \(d\) after finding the value of \(a\). 3. Coefficient of \(x^b\): \(c = 2a\), since \(b = 4\), the coefficient of \(x^4\) is \(c\). 4. Constant term: \(3 = 3a\), solving for \(a\), we get \(a = 1\). Now, substitute the value of \(a\) in the equation for \(d\): \(d + 9 = 0 \Rightarrow d = -9\).

Step 5: Writing the values of the constants

Now we can write down the possible values for the constants: \(a = 1\), \(b = 4\), \(c = 2\), and \(d = -9\).

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of multiplying two polynomials to create a new polynomial. In our given exercise, we have

• the expression: \[(3x^3 + a)(2x^b + 3)\]

which needs to be expanded to find the values of unknown variables. Expansion requires using the distributive property. We apply this property multiple times to multiply each term of the first polynomial by each term of the second polynomial. This thoroughly reveals all parts of the expanded expression. Doing this ensures that no term is left out, providing a complete and expanded view of the polynomial product.

The expansion here yields:

• (3x^3)(2x^b) = 6x^{3+b}
• (3x^3)(3) = 9x^3
• a(2x^b) = 2ax^b
• a(3) = 3a

After expanding, you are left with a polynomial where each term comes from systematically multiplying the terms in the original binomials.

Distributive Property

The distributive property is a key principle in algebra, especially when dealing with polynomial multiplication. It states that for any three numbers a, b, and c, the equation \(a(b + c) = ab + ac\) holds true. This property allows us to "distribute" one element over others linked by addition or subtraction in sums. When dealing with polynomials such as in our exercise, where you have

• two expressions: \[(3x^3 + a)\]and \[(2x^b + 3)\]

this principle guides us to multiply each term in

• \[3x^3\]by each term in \[2x^b + 3\]
• \[a\]by each term in \[2x^b + 3\]

This step-by-step application helps to ensure that every contributing factor to the polynomial is considered, and the correct results are achieved. This methodical approach is crucial in polynomial equations, leading to the precise formulation of the expanded polynomial used in further analysis.

Coefficient Comparison

Once a polynomial is expanded, as in our example, comparing coefficients allows us to equate and solve for unknowns in the polynomial expressions. Polynomial expressions have specific terms with coefficients that represent the multiplier of each variable. In the expression obtained from our expanded form,

• \[6x^{3+b} + 9x^3 + 2ax^b + 3a\]

we directly compare it with the given expression

• \[6x^7 + cx^4 + dx^3 + 3\]

by aligning coefficients of like terms. This process gives:

• For \(x^{3+b}\), we equate 6 with the first coefficient of \(x^7\) to solve for \(b = 4\).
• For \(x^3\), the comparison is \(d + 9 = 0\). Thus, \(d = -9\) after solving.
• For the constant term, equating \(3\) from the quadratic polynomial with \(3a\) gives us \(a = 1\).
• And, knowing \(b\), the coefficient \(c\) resolves to \(2\).

This method facilitates solving for unknown variables by simple equational alignment and ensures consistent polynomial structure.

Algebraic Simplification

Algebraic simplification involves reducing expressions into their simplest form, making it more straightforward to work with them in further mathematical operations. This process often follows the polynomial expansion and the distributive steps in a problem-solving sequence. Once you've completed the multiplication and coefficient comparison like in our task, what remains is to simplify terms to clear and concise expressions.

The goal is to combine and reduce terms, ensuring concise expressions without losing any necessary components or information. In this exercise, after determining that:

• \(a = 1\)
• \(b = 4\)
• \(c = 2\)
• \(d = -9\)

the rest of the coefficiency comparison should be adjusted to make it fit exactly as needed to simplify expressions coherently. This prepares the polynomial correctly, highlighting accurate solutions and validating the math throughout the equation. Simplification helps avoid errors, enabling more effortless calculations in more complex equations to come.