Problem 61 - A Graphical Approach to College Algebra

Question

Is the following factorization of \(3 a^{4}+14 a^{2}-5\) correct? Explain. If it is incorrect, give the correct factors. $$\begin{aligned}3 a^{4}+14 a^{2}-5 &=3 u^{2}+14 u-5 & & \text { Let } u=a^{2} \\\&=(3 u-1)(u+5) & & \text { Factor. }\end{aligned}$$

Step-by-Step Solution

Answer

The given factorization is correct: \((3a^2 - 1)(a^2 + 5)\).

Step 1: Identify the original expression

We are given the polynomial \(3a^4 + 14a^2 - 5\). To determine the correctness of its factorization, we need to verify each step given in the proposed factorization method.

Step 2: Make the substitution

Substitute \(u = a^2\) into the polynomial. This gives us another polynomial: \(3u^2 + 14u - 5\). The substitution is correct as we're expressing \(a^4\) as \((a^2)^2\).

Step 3: Factor the new quadratic

Now attempt to factor the quadratic expression \(3u^2 + 14u - 5\). A correct factorization would involve finding two numbers whose product is \(3 imes -5 = -15\) and whose sum is 14. These numbers are 15 and -1.

Step 4: Verify the factorization

Given the factorization \((3u - 1)(u + 5)\), expand it to check if it equals the original substituted polynomial. \((3u - 1)(u + 5) = 3u^2 + 15u - u - 5 = 3u^2 + 14u - 5\), which matches the polynomial \(3u^2 + 14u - 5\).

Step 5: Reverse the substitution

To return to the expression in terms of \(a\), replace \(u\) with \(a^2\) in the factorized form: \((3a^2 - 1)(a^2 + 5)\). This gives the complete factorization of the original polynomial \(3a^4 + 14a^2 - 5\).

Step 6: Conclusion

The proposed factorization is indeed correct. The expression \(3a^4 + 14a^2 - 5\) factors to \((3a^2 - 1)(a^2 + 5)\).

Key Concepts

Quadratic Equation

A quadratic equation is an essential tool in algebra that helps us with numerous mathematical operations. Typically, a quadratic equation has the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the variable. The structure of this equation allows for the determination of roots or solutions by using various methods, such as factoring, completing the square, or applying the quadratic formula. Quadratic equations are widespread in mathematical problems because they appear in contexts like geometry, physics, and financial calculations. Understanding how to manipulate and solve these equations lays the groundwork for more complex algebraic expressions. In our problem, we represent the polynomial \(3a^4 +14a^2 - 5\) as a quadratic equation after substitution, making it easier to handle, analyze, and factor.

Substitution Method

The substitution method is a strategic approach used to simplify complex algebraic expressions, such as the conversion of higher-degree polynomials into quadratic equations. By temporarily replacing parts of the expression with a simpler variable, we reduce complexity and focus on solving one piece at a time.In the given problem, substitution involves setting \(u = a^2\). This cleverly transforms the polynomial \(3a^4 + 14a^2 - 5\) into the more familiar form \(3u^2 + 14u - 5\), a quadratic equation.

• This conversion allows us to apply techniques meant for quadratics.
• It effectively rewires the problem-solving process, making it more manageable.

Once the calculation and factorization of the quadratic are complete, all that remains is to reverse the substitution. By this method, the solution emerges clearly and efficiently.

Expansion Verification

Verification by expansion is a vital step, ensuring factorization correctness in polynomial equations. Once a polynomial is factored, expanding the product of its factors is necessary to check if it matches the original expression.In the example exercise, the factors \((3u - 1)(u + 5)\) need to be expanded:

• Multiply \(3u - 1\) by \(u\) and \(5\) separately.
• Combine these results: \(3u^2 + 15u - u - 5\) simplifies to \(3u^2 + 14u - 5\).
• Realize that it matches the initial quadratic polynomial.

Successful verification confirms the accuracy of each factorization step and offers confidence in the correctness of the solution. Once verified, replace back the substituted variable to revert to the original polynomial format. This step adds an extra layer of assurance before finalizing the factorized expression.