Math Problem Statement

Show me an equation where you are solving for x, that involves at least one polynomial, two exponents, and in working the solution you need to find a common denominator.

Solution

Here is an equation that meets your requirements:

x2+3x+2x1=2x2+5xx+1\frac{x^2 + 3x + 2}{x - 1} = \frac{2x^2 + 5x}{x + 1}

We need to solve for xx, and the solution involves polynomials, exponents, and finding a common denominator. Here's the step-by-step solution:

Step 1: Cross multiply

To eliminate the denominators, we multiply both sides of the equation by (x1)(x+1)(x - 1)(x + 1), which is the common denominator. This will give us:

(x2+3x+2)(x+1)=(2x2+5x)(x1)(x^2 + 3x + 2)(x + 1) = (2x^2 + 5x)(x - 1)

Step 2: Expand both sides

Left-hand side:

(x2+3x+2)(x+1)=x2(x+1)+3x(x+1)+2(x+1)(x^2 + 3x + 2)(x + 1) = x^2(x + 1) + 3x(x + 1) + 2(x + 1) Expanding: =x3+x2+3x2+3x+2x+2= x^3 + x^2 + 3x^2 + 3x + 2x + 2 =x3+4x2+5x+2= x^3 + 4x^2 + 5x + 2

Right-hand side:

(2x2+5x)(x1)=2x2(x1)+5x(x1)(2x^2 + 5x)(x - 1) = 2x^2(x - 1) + 5x(x - 1) Expanding: =2x32x2+5x25x= 2x^3 - 2x^2 + 5x^2 - 5x =2x3+3x25x= 2x^3 + 3x^2 - 5x

Step 3: Set the equation equal

Now that both sides are expanded, we can set the equation:

x3+4x2+5x+2=2x3+3x25xx^3 + 4x^2 + 5x + 2 = 2x^3 + 3x^2 - 5x

Step 4: Move all terms to one side

Move all terms to the left side of the equation:

x3+4x2+5x+22x33x2+5x=0x^3 + 4x^2 + 5x + 2 - 2x^3 - 3x^2 + 5x = 0 Simplify:

x3+x2+10x+2=0-x^3 + x^2 + 10x + 2 = 0

Step 5: Solve the cubic equation

Now we need to solve for xx in the cubic equation: x3+x2+10x+2=0-x^3 + x^2 + 10x + 2 = 0

This cubic equation can be solved by methods like factoring, using the Rational Root Theorem, or numerical methods if factoring isn't feasible. Let's test potential rational roots:

By the Rational Root Theorem, the possible rational roots are ±1,±2\pm 1, \pm 2.

Testing x=1x = -1: (1)3+(1)2+10(1)+2=1+110+2=60-(-1)^3 + (-1)^2 + 10(-1) + 2 = 1 + 1 - 10 + 2 = -6 \neq 0 Testing x=2x = 2: (2)3+(2)2+10(2)+2=8+4+20+2=180-(2)^3 + (2)^2 + 10(2) + 2 = -8 + 4 + 20 + 2 = 18 \neq 0 Testing x=2x = -2: (2)3+(2)2+10(2)+2=8+420+2=60-(-2)^3 + (-2)^2 + 10(-2) + 2 = 8 + 4 - 20 + 2 = -6 \neq 0

This equation does not have simple integer roots. You may need a numerical method (e.g., Newton's method) or graphing to solve it further.

Would you like help with finding the numerical solutions or any further clarification on the steps?


Related questions to explore:

  1. How can we use synthetic division to solve cubic equations?
  2. What is the Rational Root Theorem and how does it apply to solving polynomials?
  3. How do you factor cubic polynomials by grouping?
  4. Can you explain numerical methods for solving cubic equations?
  5. What is the significance of finding a common denominator when solving equations with polynomials and fractions?

Tip: When working with polynomial equations that don't have simple integer solutions, graphing the function can often give you a good approximation of the roots.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Rational Equations
Exponents
Cubic Equations

Formulas

Cross multiplication
Rational Root Theorem
Expanding binomials

Theorems

Rational Root Theorem

Suitable Grade Level

Grades 10-12