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High School Precalculus: Help and Review32 chapters | 297 lessons

Instructor:
*David Karsner*

Quartic equations are polynomials that have a degree of four. There are several different means of solving quartic equations such as factoring, treating them like quadratics, use of a graphing calculator, and the rational root theorem.

Linear functions such as 2x-1=0 are easy to solve using inverse operations. Quadratic equations such as x2+5x+6 can be solved using the quadratic formula and breaking it down into linear factors. The polynomials of a higher order than two become more difficult to solve. Quartic equations are polynomials that have a degree of four, meaning the largest exponent is a four.

You would begin to solve quartic equations by setting it equal to zero. There will be four complex (real and imaginary) solutions, since it has a degree of four, to every quartic equation. Not every quartic equation will have four real roots. It could have 0, 1, 2, 3, or 4 real roots and imaginary roots making up the total of four. This lesson will show you several possible methods for solving quartic equations.

If the quartic equation is also a difference of squares; then it can be factored just like a difference of squares can be factored. **A difference of squares** comes in the form of a2-b2 and factors like (a+b)(a-b). For example x4-81=0 would factor like (x2+9)(x2-9)=0. The (x2-9) will factor again to (x-3)(x+3). We now have (x2+9)(x-3)(x+3)=0. This quartic has solutions at x=-3,and 3. The (x2+9) factor will never equal zero over real numbers so there will be two complex solutions.

Sometimes quartic equations can look like quadratic equations and have three terms. If a quartic has a term raised to the forth power, a term raised to the second power, and a constant; you can substitute x2 with another variable and then treat it like a quadratic.

**Quadratics** are polynomials that have a degree of two and can be solved with a variety of methods like factoring, completing the square, or using the quadratic formula.

Let's look at this example:

6x4-35x2+50

Notice it has three terms, a fourth power, a square, and a constant.

Replace x2 with the variable r.

Remeber that this means r2=x4

So we have 6r2-35r+50=0.

This quadratic will solve by factoring to give us (2r-5)(3r-10)=0.

Our solutions would be r=5/2 and 10/3.

Remember that r=x2.

That gives us x2=5/2 and x2=10/3.

One more step gives us x equal to the positive and negative square root of 5/2 and 10/3.

All four of the solutions in this quartic are real, irrational numbers.

The **rational root theorem** states if a polynomial has a rational root solution; that root will occur at x=p/q where p is a factor of the constant term and q is a factor of the leading coefficient. For example 0=2x4-7x3-11x2-15x-25. The constant term is -25, the leading coefficient is 2. The factors of -25 are positive and negative 1,5,25. The factors of 2 are positive and negative 1, and 2. The possible rational roots of this equation are positive and negative 25/2,25/1,5/1,5/2,1/1,1/2. Plug all these numbers into the original equation. The ones that equal zero are the roots or the solutions. This equation has a solution at x=5.

The real solutions to quartic equations can be found with the help of a graphing calculator. If you plug your equation into your y= bank and then hit graph; the solutions to these equations are the points at which the graph of the equation crosses the x-axis. Selecting the 2nd function button followed by the calc button will give you a menu of options. Select the zero option and it will tell you the point at which the graph and x-axis intersect. The graphing calculator will only show you the rational solutions and make an estimate of the irrational solutions. Imaginary solutions will not show on a graphing calculator.

Quartic equations can be solved using a variety of methods. Some can be solved using the difference of square factoring. Some can be made to look like quadratics and then solved like a quadratic. The rational root theorem will show where the possible roots of a equation would exist. The graphing calculator will show the real solutions to a quartic equation.

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High School Precalculus: Help and Review32 chapters | 297 lessons

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