Roots of polynomials Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Roots of polynomials questions. See exactly how to solve problems on roots-of-polynomials, vieta, quadratic, cubic.

roots-of-polynomialsvietaquadraticcubicquarticgeneral-form
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The quadratic equation 2x2−7x+3=02x^2-7x+3=0 has roots α\alpha and β\beta. Find the value of α+β\alpha+\beta.

Worked solution

  1. Identify the coefficients of the quadratic

    a=2,b=−7,c=3a=2,\quad b=-7,\quad c=3

    Read the coefficients directly from the given equation.

  2. Recall the required root-coefficient relation

    α+β=−ba\alpha+\beta=-\frac{b}{a}

    The relation follows from comparing coefficients with the factorised form.

  3. Substitute the coefficients

    α+β=−−72=72\alpha+\beta=-\frac{-7}{2}=\frac{7}{2}

    Put the numerical coefficients into the relation.

  4. State the value

    α+β=72\alpha+\beta=\frac{7}{2}

    This is the required symmetric function of the roots.

Answer
α+β=72\alpha+\beta=\frac{7}{2}
Question 2
2 markseasy
The quadratic equation 3x2+5x−2=03x^2+5x-2=0 has roots α\alpha and β\beta. Find the value of αβ\alpha\beta.

Worked solution

  1. Identify the coefficients of the quadratic

    a=3,b=5,c=−2a=3,\quad b=5,\quad c=-2

    Read the coefficients directly from the given equation.

  2. Recall the required root-coefficient relation

    αβ=ca\alpha\beta=\frac{c}{a}

    The relation follows from comparing coefficients with the factorised form.

  3. Substitute the coefficients

    αβ=−23=−23\alpha\beta=\frac{-2}{3}=-\frac{2}{3}

    Put the numerical coefficients into the relation.

  4. State the value

    αβ=−23\alpha\beta=-\frac{2}{3}

    This is the required symmetric function of the roots.

Answer
αβ=−23\alpha\beta=-\frac{2}{3}
Question 3
2 markseasy
The quadratic equation x2+6x+8=0x^2+6x+8=0 has roots α\alpha and β\beta. Find the value of α+β\alpha+\beta.

Worked solution

  1. Identify the coefficients of the quadratic

    a=1,b=6,c=8a=1,\quad b=6,\quad c=8

    Read the coefficients directly from the given equation.

  2. Recall the required root-coefficient relation

    α+β=−ba\alpha+\beta=-\frac{b}{a}

    The relation follows from comparing coefficients with the factorised form.

  3. Substitute the coefficients

    α+β=−61=−6\alpha+\beta=-\frac{6}{1}=-6

    Put the numerical coefficients into the relation.

  4. State the value

    α+β=−6\alpha+\beta=-6

    This is the required symmetric function of the roots.

Answer
α+β=−6\alpha+\beta=-6
Question 4
2 markseasy
The quadratic equation x2−3x−10=0x^2-3x-10=0 has roots α\alpha and β\beta. Find the value of αβ\alpha\beta.

Worked solution

  1. Identify the coefficients of the quadratic

    a=1,b=−3,c=−10a=1,\quad b=-3,\quad c=-10

    Read the coefficients directly from the given equation.

  2. Recall the required root-coefficient relation

    αβ=ca\alpha\beta=\frac{c}{a}

    The relation follows from comparing coefficients with the factorised form.

  3. Substitute the coefficients

    αβ=−101=−10\alpha\beta=\frac{-10}{1}=-10

    Put the numerical coefficients into the relation.

  4. State the value

    αβ=−10\alpha\beta=-10

    This is the required symmetric function of the roots.

Answer
αβ=−10\alpha\beta=-10
Question 5
2 markseasy
The cubic equation x3−4x2+x+6=0x^3-4x^2+x+6=0 has roots α\alpha, β\beta and γ\gamma. Find the value of α+β+γ\alpha+\beta+\gamma.

Worked solution

  1. Identify the coefficients of the cubic

    a=1,b=−4,c=1,d=6a=1,\quad b=-4,\quad c=1,\quad d=6

    Read the coefficients directly from the given equation.

  2. Recall the required root-coefficient relation

    α+β+γ=−ba\alpha+\beta+\gamma=-\frac{b}{a}

    The relation follows from comparing coefficients with the factorised form.

  3. Substitute the coefficients

    α+β+γ=−−41=4\alpha+\beta+\gamma=-\frac{-4}{1}=4

    Put the numerical coefficients into the relation.

  4. State the value

    α+β+γ=4\alpha+\beta+\gamma=4

    This is the required symmetric function of the roots.

Answer
α+β+γ=4\alpha+\beta+\gamma=4

Unlock 65 more Roots of polynomials questions

Create a free account to work through every Further Maths Roots of polynomials question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Roots of polynomials practice

Hard QuestionsPractice Questions

Related Pure Maths topics

Argand diagramsComplex numbers: arithmeticModelling with differential equationsComplex numbers: de MoivreFirst-order differential equationsFurther calculus