Problem 68 - A Graphical Approach to College Algebra

Question

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{4}-52 x^{2}+147 ; \quad-7 \text { and } 7$$

Step-by-Step Solution

Answer

The other roots are \(\sqrt{3}\) and \(-\sqrt{3}\).

Step 1: Recognize Given Roots

We are given that \(-7\) and \(7\) are roots of the polynomial \(P(x) = x^4 - 52x^2 + 147\). This implies that \((x + 7)\) and \((x - 7)\) are factors of \(P(x)\).

Step 2: Form the Product of Known Factors

Since \((x + 7)\) and \((x - 7)\) are factors, we can multiply them to get a quadratic factor:\[(x + 7)(x - 7) = x^2 - 49\].

Step 3: Divide Polynomial by Known Quadratic Factor

Divide \(P(x) = x^4 - 52x^2 + 147\) by \(x^2 - 49\). Perform polynomial division to do this, and solve:\[\frac{x^4 - 52x^2 + 147}{x^2 - 49}\]. The result will be another quadratic polynomial.

Step 4: Perform Polynomial Division

Through polynomial division, dividing \(x^4 - 52x^2 + 147\) by \(x^2 - 49\) results in:\[x^2 - 3\].

Step 5: Find Remaining Roots of the Quotient

Now, solve \(x^2 - 3 = 0\) for \(x\):\[x^2 = 3\]\[x = \pm\sqrt{3}\].These are the other roots of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is a method used in algebra to divide one polynomial by another, just like regular division with numbers. It helps in breaking down complex polynomials into smaller, more manageable pieces. When dividing polynomials, the goal is to express the original polynomial as the product of two or more simpler polynomials. This process can reveal factors, like roots, which are solutions to equations set to zero.

To perform polynomial division, we typically follow these steps:

• Identify the dividend (the polynomial you want to divide) and the divisor (the polynomial you are dividing by).
• Align the terms of the dividend and divisor in standard form (highest power to lowest).
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract this product from the dividend.
• Repeat the process using the result until all terms are used.

In our exercise, dividing the polynomial \(x^4 - 52x^2 + 147\) by \(x^2 - 49\) simplifies the problem, allowing us to find the other roots of the polynomial.

Quadratic Factorization

Quadratic factorization is the process of expressing a quadratic expression in the form where it is a product of two binomials. This is closely related to finding the roots or solutions of the equation.

For example, a simple quadratic, like \(x^2 - 49\), can be factored by recognizing it as a difference of squares:

• The form is \(a^2 - b^2 = (a+b)(a-b)\).
• In this form, \(x^2 - 49\) becomes \((x+7)(x-7)\).

Quadratics in different forms can often be factored by looking for patterns or by using the quadratic formula if they do not easily present themselves as products of binomials.

In our specific question, after factoring \(x^2 - 49\), we used it to further divide the original polynomial to simplify and reveal additional factors and roots, such as \(x = \pm \sqrt{3}\) from \(x^2 - 3\). This step completes the factorization of the polynomial.

Algebraic Equations

Algebraic equations are mathematical statements that use variables to represent numbers and express relationships. These equations often involve polynomial expressions. Solving these equations means finding the values of the variables that make the equations true.

The process generally involves:

• Finding the roots of the polynomial, which are the solutions to the equation.
• Applying algebraic operations such as addition, subtraction, multiplication, division, and factorization to simplify the equations.
• Using known formulas like the quadratic formula when necessary.

Understanding and solving algebraic equations are fundamental skills in mathematics, enabling us to bridge theoretical concepts with practical problems. In our initial exercise, by finding that \(x = \pm 7\) and \(x = \pm \sqrt{3}\) satisfy the polynomial \(P(x)=0\), we solve the algebraic equation \(x^4 - 52x^2 + 147=0\) completely, identifying all possible roots.