We can simplify the polynomial below because it contains a pair of like terms:
To simplify, we combine the like terms ( and ) by adding or subtracting their coefficients:
Simplify the polynomial
Here, and are like terms, so we group them, and then we add or subtract the coefficients.
$6w^2-8w^3-3w^2=$
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a
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$-8w^3+2w^2$ |
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b
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$-8w^3+3w^2$ |
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c
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$-2w^3+3w^2$ |
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d
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$-2w^3-3w^2$ |
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e
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$-8w^3-9w^2$ |
Simplify the polynomial $3b^6-b^6+7-2b^3-\dfrac{1}{2}.$
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a
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$-3b^6-2b^3+\dfrac{7}{2}$ |
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b
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$-3b^6+2b^3-\dfrac{7}{2}$ |
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c
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$4b^6-2b^3+\dfrac{15}{2}$ |
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d
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$2b^6+2b^3+\dfrac{15}{2}$ |
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e
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$2b^6-2b^3+\dfrac{13}{2}$ |
To write a polynomial in standard form, we must arrange the terms of the variable's exponent in descending order. The first term should have the largest exponent, and the last term should have the smallest exponent.
For example, consider the following polynomial:
To write this polynomial in standard form, we follow the steps below:
Identify the term with the highest exponent () and write it first.
Identify the term with the next highest exponent () and write it second.
Continue identifying the term with the next highest exponent and writing it next until all of the polynomial terms have been written.
Therefore, the standard form of our polynomial is
Note the following:
The leading term always appears first when a polynomial is written in standard form. This is where the leading term gets its name!
When a polynomial is written in standard form, we also say that the terms are written in decreasing powers of or descending powers of because the powers are decreasing/descending.
If we were to write the terms of our polynomial in increasing (or ascending) powers of it would look as follows: In this case, the powers of increase as we read from left to right.
Rewrite the polynomial below in standard form.
To write a polynomial in standard form, we must arrange the terms in descending order of the variable's exponent. The first term should have the largest exponent, and the last term should have the smallest exponent.
The highest exponent term is followed by and then and finally
So, the standard form of the polynomial is
Which of the following polynomials are given in standard form?
- $2x^3-3x^2-6$
- $6+4x-x^2$
- $-x^2+2x^3+3$
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a
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II only |
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b
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I only |
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c
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I and III |
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d
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I and II |
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e
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III only |
The polynomial $3x-2x^2+x^3$ written in standard form is
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a
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$x^3-2x^2+3x$ |
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b
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$-2x^2+x^3+3x$ |
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c
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$3x+x^3-2x^2$ |
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d
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$x^3+3x-2x^2$ |
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e
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$x^3+2x^2+3x$ |
Simplify the polynomial giving your final answer in standard form.
First, we collect like terms:
Now, we need to express our polynomial in standard form.
The highest exponent term is followed by and finally
So, the standard form of the polynomial is
Simplify the following polynomial, expressing your answer in standard form.
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a
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b
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c
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d
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e
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Simplify the following polynomial, expressing your answer in standard form.
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a
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b
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c
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d
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e
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The standard form of the polynomial $2+3x^2+4x^3-x^2-x^3$ is
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a
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$4x^3 + 3x^2 + 2$ |
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b
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$2 + 2x^2 + 3x^3$ |
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c
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$2 + 3x^2 + 4x^3$ |
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d
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$3x^3 + 2x^2 + 2$ |
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e
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$2x^2 + 3x^3$ |