How to Solve Polynomials

Co-authored by David Jia

Last Updated: September 7, 2024 Fact Checked

A polynomial is an expression made up of adding and subtracting terms. A terms can consist of constants, coefficients, and variables. When solving polynomials, you usually trying to figure out for which x-values y=0. Lower-degree polynomials will have zero, one or two real solutions, depending on whether they are linear polynomials or quadratic polynomials. These types of polynomials can be easily solved using basic algebra and factoring methods. For help solving polynomials of a higher degree, read Solve Higher Degree Polynomials .

Method 1

Method 1 of 2:

Solving a Linear Polynomial

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1
Determine whether you have a linear polynomial. A linear polynomial is a polynomial of the first degree. ^{[1]
X
Research source} This means that no variable will have an exponent greater than one. Because this is a first-degree polynomial, it will have exactly one real root, or solution. ^{[2]
X
Research source}
- For example, $5x+2$ is a linear polynomial, because the variable $x$ has no exponent (which is the same as an exponent of 1).
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2
Set the polynomial equal to zero. This is a necessary step in solving all polynomials. ^{[3]
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Research source}
- For example, $5x+2=0$
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3
Isolate the variable term. To do this, add or subtract the constant from both sides of the equation. ^{[4]
X
Expert Source

David Jia
Academic Tutor

Expert Interview. 7 January 2021.} A constant is a term without a variable. ^{[5]
X
Research source}
- For example, to isolate the $x$ term in $5x+2=0$ , you would subtract $2$ from both sides of the equation:
 $5x+2=0$
 $5x+2-2=0-2$
 $5x=-2$
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4
Solve for the variable. Usually you will need to divide each side of the equation by the coefficient. This will give you the root, or solution, to your polynomial. ^{[6]
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Research source}
- For example, to solve for $x$ in $5x=-2$ , you would divide each side of the equation by $5$ :
 $5x=-2$
 ${\frac {5x}{5}}={\frac {-2}{5}}$
 $x={\frac {-2}{5}}$
 So, the solution to $5x+2$ is $x={\frac {-2}{5}}$ .

Method 2

Method 2 of 2:

Solving a Quadratic Polynomial

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1
Determine whether you have a quadratic polynomial. A quadratic polynomial is a polynomial of the second degree. ^{[7]
X
Research source} This means that no variable will have an exponent greater than 2. Because this is a second-degree polynomial, it will have two real roots, or solutions. ^{[8]
X
Research source}
- For example, $x^{{2}}+8x-20$ is a quadratic polynomial, because the variable $x$ has an exponent of $2$ .
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2
Make sure the polynomial is written in order of degree. This means that the term with the exponent of $2$ is listed first, followed by the first-degree term, followed by the constant. ^{[9]
X
Research source}
- For example, you would rewrite $8x+x^{{2}}-20$ as $x^{{2}}+8x-20$ .
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3
Set the equation to equal zero. This is a necessary step for solving all polynomials. ^{[10]
X
Research source}
- For example, $x^{{2}}+8x-20=0$ .
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4
Rewrite the expression as a four-term expression. To do this, split up the first-degree term (the $x$ term). You are looking for two numbers whose sum is equal to the first degree coefficient, and whose product is equal to the constant. ^{[11]
X
Research source}
- For example, for the quadratic polynomial $x^{{2}}+8x-20=0$ , you need to find two numbers ( $a$ and $b$ ), where $a+b=8$ , and $a\cdot b=-20$ .
- Since you have $-20$ , you know that one of the number will be negative.
- You should see that $10+(-2)=8$ and $10\cdot (-2)=-20$ . Thus, you will split up $8x$ into $10x-2x$ and rewrite the quadratic polynomial: $x^{{2}}+10x-2x-20=0$ .
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5
Factor by grouping. To do this, factor out a term common to the first two terms in the polynomial. ^{[12]
X
Research source}
- For example, the first two terms in the polynomial $x^{{2}}+10x-2x-20=0$ are $x^{{2}}+10x$ . A term common to both is $x$ . Thus, the factored group is $x(x+10)$ .
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6
Factor the second group. To do this, factor out a term common to the second two terms in the polynomial. ^{[13]
X
Research source}
- For example, the second two terms in the polynomial $x^{{2}}+10x-2x-20=0$ are $-2x-20$ . A term common to both is $-2$ . Thus, the factored group is $-2(x+10)$ .
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7
Rewrite the polynomial as two binomials. A binomial is a two-term expression. You already have one binomial, which is the expression in parentheses for each group. This expression should be the same for each group. The second binomial is created by combining the two terms that were factored out of each group. ^{[14]
X
Research source}
- For example, after factoring by grouping, $x^{{2}}+10x-2x-20=0$ becomes $x(x+10)-2(x+10)=0$ .
- The first binomial is $(x+10)$ .
- The second binomial is $(x-2)$ .
- So the original quadratic polynomial, $x^{{2}}+8x-20=0$ can be written as the factored expression $(x+10)(x-2)=0$ .
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8
Find the first root, or solution. To do this, solve for $x$ in the first binomial. ^{[15]
X
Research source}
- For example, to find the first root for $(x+10)(x-2)=0$ , you would first set the first binomial expression to $0$ and solve for $x$ . Thus:
 $x+10=0$
 $x+10-10=0-10$
 $x=-10$
 So, the first root of the quadratic polynomial $x^{{2}}+8x-20=0$ is $-10$ .
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9
Find the second root, or solution. To do this, solve for $x$ in the second binomial. ^{[16]
X
Research source}
- For example, to find the second root for $(x+10)(x-2)=0$ , you would set the second binomial expression to $0$ and solve for $x$ . Thus:
 $x-2=0$
 $x-2+2=0+2$
 $x=2$
 So, the second root of the quadratic polynomial $x^{{2}}+8x-20=0$ is $2$ .

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How did you get -2 in the second binomial?

Donagan

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The original equation was 5x + 2 = 0. Then 2 was subtracted from both sides of the equation in order to begin the process of solving for x. This resulted in 5x = -2.

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What do I get if I add x - 2 and 1/x?

Donagan

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You get x - 2 + 1/x.

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For trinomials, would I turn them into a quadratic polynomials and then binomials?

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Yes. To factor a trinomial, you must split it into a quadratic polynomial.

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Remember the order of operations while you work -- First work in the parenthesis, then do the multiplication and division, and finally do the addition and subtraction. ^{[17]
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Don't fret if you get different variables, like t, or if you see an equation set to f(x) instead of 0. If the question wants roots, zeros, or factors, just treat it like any other problem.

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How to Solve Polynomials

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Solving a Linear Polynomial

Solving a Quadratic Polynomial

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