Exams · Russian state exam, grade 9
No. 20 Fractions and factoring
Reduce a fraction or factor a polynomial.
How to solve it
Reducing fractions
To reduce a fraction like $\frac{x^2 + 7x + 6}{x^2 + 5x + 4}$ is to divide its numerator and denominator by their common factor. Only factors can be cancelled, so both polynomials are factored first.
Step by step
- Factor the numerator: $x^2 + 7x + 6 = (x + 1)(x + 6)$.
- Factor the denominator: $x^2 + 5x + 4 = (x + 1)(x + 4)$.
- Cancel the common factor: you get $\frac{x + 6}{x + 4}$.
- Remember where the original fraction is undefined: the reduced one equals it everywhere except at these points (here $x \ne -1$ and $x \ne -4$).
Common mistakes
- Cancelling terms instead of factors: nothing cancels in $\frac{x + 6}{x + 4}$.
- Not factoring completely and missing the common factor.
- Missing factors that differ only in sign: $1 - x = -(x - 1)$.
Example
Factoring
To factor a polynomial is to write it as a product of simpler ones: $x^2 - 5x + 6 = (x - 2)(x - 3)$. You need it on its own, to reduce fractions and to solve equations.
Step by step
- Take out a common factor if there is one: $2x^2 - 6x = 2x(x - 3)$.
- Look for the special products, for instance $a^2 - b^2 = (a - b)(a + b)$.
- A quadratic factors through its roots.
- For a cubic, find an integer root among the divisors of the constant term, check it by substituting, and divide the polynomial by $(x - x_0)$.
- Go on until every factor is linear or cannot be factored further.
Common mistakes
- Losing the leading coefficient: $2x^2 - 10x + 12 = 2(x - 2)(x - 3)$, and without the $2$ the equality is false.
- Getting the signs wrong: the root $2$ gives the factor $(x - 2)$, not $(x + 2)$.
- Stopping halfway, when one of the factors still splits.