1Cademy - Subtracting $$\frac{4x^2-11x+8}{x^2-3x+2}

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How to Add and Subtract Rational Expressions with a Common Denominator

Example

Subtracting $\frac{4x^2-11x+8}{x^2-3x+2} - \frac{3x^2+x-3}{x^2-3x+2}$

Subtract two rational expressions that share the common denominator $x^2-3x+2$ :

$\frac{4x^2-11x+8}{x^2-3x+2} - \frac{3x^2+x-3}{x^2-3x+2}$

Step 1 — Subtract the numerators over the common denominator. Place the second numerator in parentheses and write the difference over the shared denominator:

$\frac{4x^2-11x+8 - (3x^2+x-3)}{x^2-3x+2}$

Step 2 — Distribute the negative sign in the numerator. Multiply each term inside the parentheses by $-1$ , changing $+3x^2$ to $-3x^2$ , $+x$ to $-x$ , and $-3$ to $+3$ :

$\frac{4x^2-11x+8 - 3x^2 - x + 3}{x^2-3x+2}$

Step 3 — Combine like terms in the numerator. Group the $x^2$ -terms: $4x^2 - 3x^2 = x^2$ . Group the $x$ -terms: $-11x - x = -12x$ . Group the constants: $8 + 3 = 11$ :

$\frac{x^2-12x+11}{x^2-3x+2}$

Step 4 — Factor both the numerator and the denominator. The numerator $x^2-12x+11$ factors as $(x-1)(x-11)$ , since $(-1) + (-11) = -12$ and $(-1)(-11) = 11$ . The denominator $x^2-3x+2$ factors as $(x-1)(x-2)$ , since $(-1) + (-2) = -3$ and $(-1)(-2) = 2$ :

$\frac{(x-1)(x-11)}{(x-1)(x-2)}$

Step 5 — Simplify by removing the common factor. Cancel the shared factor $(x-1)$ from the numerator and denominator:

$\frac{x-11}{x-2}$

This example reinforces the procedure of subtracting rational expressions with a common denominator, distributing the negative sign across a trinomial numerator, and factoring both the resulting numerator and the common denominator to identify and cancel common factors.

Updated 2026-04-30

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OpenStax Intermediate Algebra

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