Solve $2x^4 - 5x^3 - 7x^2 + 10x + 6 = 0$. Give exact answers.

$x = 3, -\frac{1}{2}, \sqrt{2}, -\sqrt{2}$

Grade 12 · MHF4U · Advanced Functions

MHF4U: Solving Polynomial Equations

A polynomial equation is solved by factoring. Get zero on one side, break the polynomial into factors, and set each factor equal to zero. The roots are the $x$ -intercepts of the graph, so a sketch tells you how many to expect.

How It Works

First, move everything to one side so the equation reads $P(x) = 0$ . A product is zero only when one of its factors is, which is the whole method; it does not work with a $5$ on the right.

Then factor, in this order. Take out a common factor, including a plain $x$ . Look for grouping in a four-term cubic. Look for a difference of squares or a sum or difference of cubes. If none of those works, use the factor theorem: test the rational candidates $\pm\frac{p}{q}$ , with $p$ dividing the constant term and $q$ the leading coefficient, until $P(a) = 0$ , then divide by $x - a$ .

Each division drops the degree by one. Stop when what is left is a quadratic, and factor it or use the quadratic formula. A quadratic factor with a negative discriminant has no real zeros, so it contributes no roots.

A degree $n$ equation has at most $n$ real roots. Fewer is normal: a repeated factor counts once, and an irreducible quadratic contributes none. A quartic can have four, two or zero real roots, and a cubic always has at least one.

Check by substituting each root back, or by looking at where the graph crosses the axis. And write out every root, including $0$ when $x$ was the common factor; it is the one students most often drop.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Solve $x^3 - 9x = 0$ .
Show solution
1. Common factor: $x(x^2 - 9) = 0$ .
2. Difference of squares: $x(x - 3)(x + 3) = 0$ .
3. Each factor to zero: $x = 0$ , $x = 3$ , $x = -3$ .
Answer: $x = -3, 0, 3$
Easy
Solve $x^3 - 3x^2 - 4x + 12 = 0$ .
Show solution
1. Group: $x^2(x - 3) - 4(x - 3) = 0$ .
2. Common factor: $(x - 3)(x^2 - 4) = 0$ , then $(x - 3)(x - 2)(x + 2) = 0$ .
Answer: $x = -2, 2, 3$
Medium
Solve $x^3 - 4x^2 + x + 6 = 0$ .
Show solution
1. Grouping fails, so use the factor theorem. Candidates are factors of $6$ : $\pm 1, \pm 2, \pm 3, \pm 6$ .
2. $P(1) = 1 - 4 + 1 + 6 = 4$ . $P(-1) = -1 - 4 - 1 + 6 = 0$ , so $x + 1$ is a factor.
3. Divide by $x + 1$ : the quotient is $x^2 - 5x + 6 = (x - 2)(x - 3)$ .
4. $(x + 1)(x - 2)(x - 3) = 0$ .
Answer: $x = -1, 2, 3$
Medium
Solve $2x^3 + 3x^2 - 8x + 3 = 0$ .
Show solution
1. Leading coefficient $2$ and constant $3$ : candidates are $\pm 1, \pm 3, \pm\frac{1}{2}, \pm\frac{3}{2}$ .
2. $P(1) = 2 + 3 - 8 + 3 = 0$ , so $x - 1$ is a factor.
3. Divide by $x - 1$ : $2x^2 + 5x - 3$ , which factors as $(2x - 1)(x + 3)$ .
4. $(x - 1)(2x - 1)(x + 3) = 0$ .
Answer: $x = -3, \frac{1}{2}, 1$
Test level
Solve $2x^4 - 5x^3 - 7x^2 + 10x + 6 = 0$ . Give exact answers.
Show solution
1. Candidates: $\pm 1, \pm 2, \pm 3, \pm 6, \pm\frac{1}{2}, \pm\frac{3}{2}$ . $P(1) = 6$ and $P(-1) = -4$ , so try $P(3) = 162 - 135 - 63 + 30 + 6 = 0$ . $x - 3$ is a factor.
2. Synthetic division by $3$ on $2, -5, -7, 10, 6$ gives $2, 1, -4, -2$ with remainder $0$ : the quotient is $2x^3 + x^2 - 4x - 2$ .
3. Group the cubic: $x^2(2x + 1) - 2(2x + 1) = (2x + 1)(x^2 - 2)$ .
4. $(x - 3)(2x + 1)(x^2 - 2) = 0$ . The last factor gives $x^2 = 2$ , so $x = \pm\sqrt{2}$ .
Answer: $x = 3, -\frac{1}{2}, \sqrt{2}, -\sqrt{2}$

Learn It on Martian Lab

Martian Lab teaches this in the Factor Theorem and Rational Zeros lesson of its MHF4U Polynomial Equations unit. Watch it, then take the Polynomial Equations practice tests to see if it stuck.

Watch the Factor Theorem and Rational Zeros lesson free

Keep Going

Free Polynomial Equations practice test, with solutions

More Polynomial Equations topics: Long and Synthetic Division · Remainder and Factor Theorem · Polynomial Inequalities · Sum and Difference of Cubes

More MHF4U topics: Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations · Laws of Logarithms · Solving Logarithmic Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Solving Exponential Equations · Graphing Logarithmic Functions · Exponential and Log Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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