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Solve: ( x + 5 ) 3 2 = 8.

{ −1 }

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Solving equations using factoring

We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.

Polynomial equations

A polynomial of degree n is an expression of the type

a n x n + a n 1 x n 1 + + a 2 x 2 + a 1 x + a 0

where n is a positive integer and a n , , a 0 are real numbers and a n 0.

Setting the polynomial equal to zero gives a polynomial equation    . The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent n .

Solving a polynomial by factoring

Solve the polynomial by factoring: 5 x 4 = 80 x 2 .

First, set the equation equal to zero. Then factor out what is common to both terms, the GCF.

5 x 4 80 x 2 = 0 5 x 2 ( x 2 16 ) = 0

Notice that we have the difference of squares in the factor x 2 16 , which we will continue to factor and obtain two solutions. The first term, 5 x 2 , generates, technically, two solutions as the exponent is 2, but they are the same solution.

5 x 2 = 0 x = 0 x 2 16 = 0 ( x 4 ) ( x + 4 ) = 0 x = 4 x = −4

The solutions are 0  (double solution), 4 , and −4.

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Solve by factoring: 12 x 4 = 3 x 2 .

x = 0 , x = 1 2 , x = 1 2

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Solve a polynomial by grouping

Solve a polynomial by grouping: x 3 + x 2 9 x 9 = 0.

This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested.

x 3 + x 2 9 x 9 = 0 x 2 ( x + 1 ) 9 ( x + 1 ) = 0 ( x 2 9 ) ( x + 1 ) = 0

The grouping process ends here, as we can factor x 2 9 using the difference of squares formula.

( x 2 9 ) ( x + 1 ) = 0 ( x 3 ) ( x + 3 ) ( x + 1 ) = 0 x = 3 x = −3 x = −1

The solutions are 3 , −3 , and −1. Note that the highest exponent is 3 and we obtained 3 solutions. We can see the solutions, the x- intercepts, on the graph in [link] .

Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).
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Solving radical equations

Radical equations are equations that contain variables in the radicand    (the expression under a radical symbol), such as

3 x + 18 = x x + 3 = x 3 x + 5 x 3 = 2

Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions    , roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.

Radical equations

An equation containing terms with a variable in the radicand is called a radical equation    .

Given a radical equation, solve it.

  1. Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.
  2. If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an n th root radical, raise both sides to the n th power. Doing so eliminates the radical symbol.
  3. Solve the remaining equation.
  4. If a radical term still remains, repeat steps 1–2.
  5. Confirm solutions by substituting them into the original equation.

Questions & Answers

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
2x + 7 =19
2x +7=19. 2x=19 - 7 2x=12 x=6
because x is 6
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
4x=3-2 4x=1 x=1+4 x=5 5x
hi can you give another equation I'd like to solve it
what is the value of x in 4x-2+3
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
then x=-1/4
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
can you help me on this topic of Geometry if l help you
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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