Algebra 2 (HS pathway)

Rational and Radical Functions

Algebra 2

✓By the end of this lesson students will be able to simplify rational expressions by factoring and canceling common factors.
✓By the end of this lesson students will be able to perform operations (multiplication, division, addition, subtraction) on rational expressions.
✓By the end of this lesson students will be able to solve rational equations and identify extraneous solutions.
✓By the end of this lesson students will be able to solve radical equations involving square roots and cube roots.
✓By the end of this lesson students will be able to check for extraneous solutions when solving radical equations.

Key concepts

Rational Expression

A rational expression is a fraction where the numerator and denominator are polynomials. The denominator cannot be zero. The domain of a rational expression is all real numbers for which the denominator is not equal to zero.

P(x)/Q(x), where Q(x) ≠ 0

Simplifying Rational Expressions

To simplify a rational expression, factor both the numerator and the denominator completely. Then, cancel out any common factors that appear in both the numerator and the denominator. Remember to state any restrictions on the variable that make the original denominator zero.

Rational Equation

A rational equation is an equation that contains one or more rational expressions. To solve a rational equation, find a common denominator for all terms, multiply both sides of the equation by the least common denominator (LCD) to eliminate the denominators, and then solve the resulting polynomial equation. Always check for extraneous solutions by substituting the solutions back into the original equation to ensure they do not make any denominator zero.

Radical Expression

A radical expression is an expression that contains a square root, cube root, or other root. The symbol √ is called the radical symbol. The number under the radical symbol is called the radicand. For even roots (like square roots), the radicand must be non-negative for the expression to be a real number.

ⁿ√x

Radical Equation

A radical equation is an equation in which the variable appears under a radical sign. To solve a radical equation, isolate the radical term on one side of the equation, then raise both sides of the equation to the power equal to the index of the radical to eliminate the radical. Solve the resulting equation. It is crucial to check all solutions in the original equation, as extraneous solutions can arise from squaring or raising both sides to an even power.

Key facts to remember

1The domain of a rational expression excludes values that make the denominator zero.
2To simplify rational expressions, factor the numerator and denominator and cancel common factors.
3When adding or subtracting rational expressions, find a common denominator.
4To solve rational equations, multiply by the LCD to clear denominators, then solve the resulting equation.
5Always check for extraneous solutions in rational equations by ensuring they don't make any original denominator zero.
6To solve radical equations, isolate the radical, then raise both sides to the power of the radical's index.
7Always check for extraneous solutions in radical equations by substituting them back into the original equation.
8For even roots, the radicand must be non-negative to yield a real number.

Worked examples

Example 1

Simplify the rational expression: (x^2 - 4x - 12) / (x^2 - 4)

IFactor the numerator: x^2 - 4x - 12 = (x - 6)(x + 2)

IIFactor the denominator: x^2 - 4 = (x - 2)(x + 2)

IIIRewrite the expression with factored terms: ((x - 6)(x + 2)) / ((x - 2)(x + 2))

IVIdentify restrictions: The denominator cannot be zero, so x ≠ 2 and x ≠ -2.

VCancel the common factor (x + 2): (x - 6) / (x - 2)

Answer

(x - 6) / (x - 2), where x ≠ 2 and x ≠ -2

It's important to state the restrictions on the variable from the original expression.

Example 2

Solve the equation: 3/(x+2) + 1/(x-2) = 4x/(x^2-4)

IFactor the denominator x^2 - 4 to (x - 2)(x + 2).

IIIdentify the LCD: (x - 2)(x + 2).

IIIIdentify restrictions: x ≠ 2 and x ≠ -2.

IVMultiply every term by the LCD: ((x - 2)(x + 2)) * (3/(x+2)) + ((x - 2)(x + 2)) * (1/(x-2)) = ((x - 2)(x + 2)) * (4x/((x-2)(x+2)))

VSimplify: 3(x - 2) + 1(x + 2) = 4x

VIDistribute: 3x - 6 + x + 2 = 4x

VIICombine like terms: 4x - 4 = 4x

VIIISubtract 4x from both sides: -4 = 0

9This is a false statement, indicating there is no solution.

Answer

No solution.

Sometimes, rational equations lead to no solutions or extraneous solutions.

Example 3

Solve the equation: √(2x + 7) - x = 2

IIsolate the radical: √(2x + 7) = x + 2

IISquare both sides: (√(2x + 7))^2 = (x + 2)^2

IIISimplify: 2x + 7 = x^2 + 4x + 4

IVRearrange into a quadratic equation (set to zero): 0 = x^2 + 2x - 3

VFactor the quadratic: 0 = (x + 3)(x - 1)

VISolve for x: x = -3 or x = 1

VIICheck solutions in the original equation:

VIIIFor x = -3: √(2(-3) + 7) - (-3) = 2 => √(-6 + 7) + 3 = 2 => √(1) + 3 = 2 => 1 + 3 = 2 => 4 = 2 (False, so x = -3 is an extraneous solution)

9For x = 1: √(2(1) + 7) - 1 = 2 => √(2 + 7) - 1 = 2 => √(9) - 1 = 2 => 3 - 1 = 2 => 2 = 2 (True, so x = 1 is a valid solution)

Answer

x = 1

Always check solutions for radical equations, especially when squaring both sides, as extraneous solutions are common.

Common mistakes

✗Canceling terms that are not factors in rational expressions (e.g., incorrectly simplifying (x+1)/x to 1).
✗Forgetting to check for extraneous solutions in rational and radical equations.
✗Not distributing the negative sign correctly when subtracting rational expressions.
✗Squaring only part of an expression when solving radical equations (e.g., incorrectly expanding (x+2)^2 as x^2+4 instead of x^2+4x+4).
✗Assuming that √(x^2) is always x instead of |x| (though for solving equations like x^2=k, x=±√k is appropriate).

Exam tips

★Always factor completely before simplifying rational expressions.
★When solving equations, clearly state any restrictions on the variable at the beginning.
★Show all steps when checking for extraneous solutions; this can earn partial credit even if the initial solution is incorrect.
★Use parentheses carefully, especially when multiplying by the LCD or squaring binomials.

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