Algebraic and partial fractions Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Algebraic and partial fractions questions. See exactly how to solve problems on algebraic-fractions, simplify, combine, partial-fractions.

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A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Simplify x2+5x+6x2+x−2\frac{x^{2} + 5 x + 6}{x^{2} + x - 2}.

Worked solution

  1. Write the fraction

    x2+5x+6x2+x−2\frac{x^{2} + 5 x + 6}{x^{2} + x - 2}

    Start from the given algebraic fraction.

  2. Factorise and cancel the common factor

    (x+2)(x+3)(x−1)(x+2)=x+3x−1\frac{\left(x + 2\right) \left(x + 3\right)}{\left(x - 1\right) \left(x + 2\right)}=\frac{x + 3}{x - 1}

    Cancel the factor common to numerator and denominator.

  3. State the simplified fraction

    x+3x−1\frac{x + 3}{x - 1}

    This is the fraction in its simplest form.

Answer
x+3x−1\frac{x + 3}{x - 1}
Question 2
2 markseasy
Simplify x2−3x−4x2+x−20\frac{x^{2} - 3 x - 4}{x^{2} + x - 20}.

Worked solution

  1. Write the fraction

    x2−3x−4x2+x−20\frac{x^{2} - 3 x - 4}{x^{2} + x - 20}

    Start from the given algebraic fraction.

  2. Factorise and cancel the common factor

    (x−4)(x+1)(x−4)(x+5)=x+1x+5\frac{\left(x - 4\right) \left(x + 1\right)}{\left(x - 4\right) \left(x + 5\right)}=\frac{x + 1}{x + 5}

    Cancel the factor common to numerator and denominator.

  3. State the simplified fraction

    x+1x+5\frac{x + 1}{x + 5}

    This is the fraction in its simplest form.

Answer
x+1x+5\frac{x + 1}{x + 5}
Question 3
2 markseasy
Simplify x2−x−6x2+6x+8\frac{x^{2} - x - 6}{x^{2} + 6 x + 8}.

Worked solution

  1. Write the fraction

    x2−x−6x2+6x+8\frac{x^{2} - x - 6}{x^{2} + 6 x + 8}

    Start from the given algebraic fraction.

  2. Factorise and cancel the common factor

    (x−3)(x+2)(x+2)(x+4)=x−3x+4\frac{\left(x - 3\right) \left(x + 2\right)}{\left(x + 2\right) \left(x + 4\right)}=\frac{x - 3}{x + 4}

    Cancel the factor common to numerator and denominator.

  3. State the simplified fraction

    x−3x+4\frac{x - 3}{x + 4}

    This is the fraction in its simplest form.

Answer
x−3x+4\frac{x - 3}{x + 4}
Question 4
2 markseasy
Simplify x2+3x−10x2−8x+12\frac{x^{2} + 3 x - 10}{x^{2} - 8 x + 12}.

Worked solution

  1. Write the fraction

    x2+3x−10x2−8x+12\frac{x^{2} + 3 x - 10}{x^{2} - 8 x + 12}

    Start from the given algebraic fraction.

  2. Factorise and cancel the common factor

    (x−2)(x+5)(x−6)(x−2)=x+5x−6\frac{\left(x - 2\right) \left(x + 5\right)}{\left(x - 6\right) \left(x - 2\right)}=\frac{x + 5}{x - 6}

    Cancel the factor common to numerator and denominator.

  3. State the simplified fraction

    x+5x−6\frac{x + 5}{x - 6}

    This is the fraction in its simplest form.

Answer
x+5x−6\frac{x + 5}{x - 6}
Question 5
2 markseasy
Simplify x2−8x+7x2−4x−21\frac{x^{2} - 8 x + 7}{x^{2} - 4 x - 21}.

Worked solution

  1. Write the fraction

    x2−8x+7x2−4x−21\frac{x^{2} - 8 x + 7}{x^{2} - 4 x - 21}

    Start from the given algebraic fraction.

  2. Factorise and cancel the common factor

    (x−7)(x−1)(x−7)(x+3)=x−1x+3\frac{\left(x - 7\right) \left(x - 1\right)}{\left(x - 7\right) \left(x + 3\right)}=\frac{x - 1}{x + 3}

    Cancel the factor common to numerator and denominator.

  3. State the simplified fraction

    x−1x+3\frac{x - 1}{x + 3}

    This is the fraction in its simplest form.

Answer
x−1x+3\frac{x - 1}{x + 3}

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