Problem 14 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 18 a^{2}-4 a, 2 a $$

Step-by-Step Solution

Answer

Based on the step-by-step solution provided above, the other factor is: $$8x^3 - 3x^2 + 4x + 5$$

Step 1: Set Up Long Division

Set up long division for the given expression divided by the known factor. $$ \require{enclose} \begin{array}{c|cccc} \multicolumn{2}{r}{8x^3}& -3x^2 & +4x &+5\\ \cline{2-5} 11x & 88x^4 & -33x^3 & +44x^2 & +55x \\ \cline{2-2} \end{array} $$

Step 2: Divide the First Term

Divide the first term of the given expression, \(88x^4\), by the first term of the known factor, \(11x\). $$ \frac{88x^4}{11x} = 8x^3 $$

Step 3: Multiply and Subtract

Multiply the known factor by the previous result and subtract it from the given expression. $$ \begin{array}{c|cccc} \multicolumn{2}{r}{8x^3}& -3x^2 & +4x &+5 \\ \cline{2-5} 11x & 88x^4 & -33x^3 & +44x^2 & +55x \\ \cline{2-2} \multicolumn{2}{r}{88x^4} & -33x^3\\ \cline{2-3} \multicolumn{2}{r}{0}&0x^2&+44x^2\\ \cline{3-3} \end{array} $$

Step 4: Continue Dividing, Multiplying, and Subtracting

Continue with the same process: we divide the next term by the first term of the known factor, then multiply and subtract. $$ \begin{array}{c|cccc} \multicolumn{2}{r}{8x^3}& -3x^2 & +4x &+5 \\ \cline{2-5} 11x & 88x^4 & -33x^3 & +44x^2 & +55x \\ \cline{2-2} \multicolumn{2}{r}{88x^4} & -33x^3\\ \cline{2-3} \multicolumn{2}{r}{0}&0x^2&+44x^2\\ \multicolumn{3}{r}{0}&-44x^2&\\ \cline{4-4} \multicolumn{2}{r}{0}&0 &0 &+55x\\ \multicolumn{4}{r}{0} & 0&+55x\\ \cline{5-5} \multicolumn{2}{r}{0}& & &0 \\ \end{array} $$

Step 5: Write the Other Factor

The quotient is the other factor. So the other factor is: $$ 8x^3-3x^2+4x+5 $$

Key Concepts

Algebraic Expressions

Algebraic expressions form the backbone of algebra and consist of variables, coefficients, and constants combined using mathematical operations such as addition, subtraction, and multiplication. In this exercise, we dealt with the algebraic expression \(88x^4 - 33x^3 + 44x^2 + 55x\). This expression includes:

• Variables: These are symbols, often \(x\), representing numbers in an expression.
• Coefficients: Numbers multiplying the variables, like 88, -33, 44, and 55 in our example.
• Terms: Each part of the expression separated by a plus or minus sign. Here, the terms are \(88x^4\), \(-33x^3\), \(+44x^2\), and \(+55x\).
• Exponent: Indicates how many times to multiply the variable by itself. For instance, \( x^4 \) means \( x \times x \times x \times x \).

Algebraic expressions allow us to solve complex problems by representing numbers and operations symbolically. Breaking down these expressions helps in understanding and solving them systematically.

Long Division Method

The long division method is an essential technique in algebra for dividing polynomials, similar to how we divide numbers. It involves several systematic steps to divide the dividend (main polynomial) by the divisor (the known factor). Here's how it works:

• Set Up the Division: Write the dividend inside a division bracket and the divisor outside, just like in number division.
• Divide the Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. This quotient becomes the first term of the answer.
• Multiply and Subtract: Multiply the entire divisor by this new term and subtract the result from the original polynomial. Repeat with the new polynomial remaining.

In the exercise, we divided \(88x^4 - 33x^3 + 44x^2 + 55x\) by \(11x\) using this method. We carefully went through each step, ensuring each term was correctly divided and subtracted, ultimately finding the other factor. The process is repeated until the remaining polynomial cannot be divided by the divisor. The remainder, in some cases, can be zero as seen here.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, they are sums of terms of the form \( ax^n \), where \( a \) is a coefficient, and \( n \) is a non-negative integer exponent.

• Degrees and Terms: A polynomial's degree is the highest power of the variable in the expression. For our exercise, the degree is 4, indicated by \( x^4 \), making it a fourth-degree polynomial.
• Classification:
  • Monomial: A single term (e.g., \( 5x^3 \)).
  • Binomial: Two terms (e.g., \( 3x^2 + 2x \)).
  • Trinomial: Three terms (e.g., \( x^2 + x + 1 \)).
• Operations: Unlike other expressions, polynomials can be added, subtracted, and multiplied easily, but division often requires methods like long division.

Understanding polynomials and their structure is critical as they are used in various mathematical models and real-world applications. They simplify complex problems by representing them in a structured and manageable form.