Problem 18 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &(x+4)(x-3)\\\ &x+4\\\ &x-3 \end{aligned}$$

Step-by-Step Solution

Answer

The lowest common multiple (LCM) of the polynomials \( (x+4)(x-3) \), \( x+4 \), and \( x-3 \) is \( x^2 + x -12 \).

Step 1: Identify the highest degree polynomial

Identify and represent the highest degree polynomial among the given polynomials, which is \( (x+4)(x-3) = x^2 + x - 12 \).

Step 2: Test divisibility

Check to see if the remaining given polynomials, \( x+4 \) and \( x-3 \), divide the polynomial identified in step one evenly. This is done by performing polynomial division.

Step 3: Conclusion

Since \( x^2 + x -12 \) is divisible evenly by both \( x+4 \) and \( x-3 \), \( x^2 + x -12 \) is the least common multiple of the given polynomials.

Key Concepts

Polynomial Division

Polynomial division is a fundamental process used to determine if one polynomial is a factor of another. It is similar to long division with numbers, allowing you to divide a polynomial by another polynomial.

When dividing polynomials, start by arranging them in standard form, with terms in descending order of degree. For our example, if you're dividing \( (x^2 + x - 12) \) by \( x+4 \), first write it as:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result above and subtract from the original polynomial.
• Bring down the next term from the dividend and repeat until there are no terms left.

Performing this division will reveal if the divisor \( x+4 \) is a factor of \( x^2 + x - 12 \). If no remainder is left, it divides evenly. The same steps apply for dividing by \( x-3 \).

Polynomial division confirms if the polynomial \( x^2 + x - 12 \) is perfectly divisible by both \( x+4 \) and \( x-3 \).

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied, give back the original polynomial.

For the polynomial \( x^2 + x - 12 \), notice it can be decomposed into the product of its factors, which are \( (x+4) \) and \( (x-3) \). This is the reverse of expanding polynomials, where multiplication of these factors yields the original polynomial expression.

To factor a polynomial:

• Look for two numbers that multiply to give the constant term, in this case, \(-12\), and add to give the linear coefficient, here \(1\).
• For \( x^2 + x - 12 \), these numbers are \(4\) and \(-3\).
• Rewrite the polynomial as \((x+4)(x-3)\).

Factoring is useful for identifying the Least Common Multiple (LCM) of polynomials because it helps verify if each given polynomial is a factor of the LCM.

Polynomial Multiplication

Polynomial multiplication is a method used to expand expressions into a standard polynomial form. It involves combining like terms after performing the distributive properties.
In the context of LCM for polynomials, start by verifying that multiplying the polynomial factors \((x+4)\) and \((x-3)\) gives the expression \(x^2 + x - 12\). This confirms the expansion is correct and embodies all factors.

• Distribute each term in one polynomial to every term in the other polynomial.
• Combine like terms to simplify the expression.

For example:\[(x+4)(x-3) = x(x-3) + 4(x-3) = x^2 - 3x + 4x - 12\]After combining like terms, it results in \(x^2 + x - 12\).

Polynomial multiplication helps in simplifying expressions to reach the LCM, ensuring that it divides evenly by the factors \(x+4\) and \(x-3\).