Section 1
📘 Dividing Polynomials Using Long Division
New Concept
The result can be expressed as:
What’s next
Next, you’ll apply this method to divide various polynomials and test if one polynomial is a factor of another.
In this Grade 10 Saxon Algebra 2 lesson, students learn how to divide polynomials using long division, applying the same dividend-divisor-quotient-remainder structure as integer long division to expressions involving monomials, linear polynomials, and higher-degree polynomials. Students also use polynomial long division to determine whether one polynomial is a factor of another by checking for a zero remainder. The lesson includes real-world application problems such as finding the ratio of volume to surface area for rectangular prisms with polynomial dimensions.
Section 1
📘 Dividing Polynomials Using Long Division
The result can be expressed as:
Next, you’ll apply this method to divide various polynomials and test if one polynomial is a factor of another.
Section 2
Polynomial long division
Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree. The final result is expressed in the form:
To divide by , we perform long division to find the quotient is with a remainder of .
To divide by , the result is a quotient of with a remainder of .
When dividing by , the quotient is with a remainder of , written as .
Think of this as the algebraic version of dividing 125 by 5. You divide the leading terms, multiply the result by the divisor, subtract the product, and bring down the next term. This process repeats until you have a remainder that is of a lower degree than the divisor, systematically simplifying complex polynomial fractions.
Section 3
Don't forget the zero placeholder
When a dividend is missing a term for a specific power of the variable, you must insert a placeholder term with a coefficient of 0. For example, to divide , you should rewrite it as .
To divide by , rewrite the dividend as to keep columns aligned.
To divide by , rewrite it as before starting the division.
Imagine lining up soldiers by rank. If a rank is missing, you leave a space for it, right? Using a zero placeholder like '' does the same thing. It keeps all your terms aligned correctly during the subtraction steps of long division, preventing you from mixing up different powers of and getting a chaotic result.
Section 4
Is it a factor? Check the remainder!
A polynomial, such as , is considered a factor of another polynomial if the remainder is 0 after division. If the remainder is any non-zero value, then it is not a factor.
Is a factor of ? Yes, because dividing gives a remainder of .
Is a factor of ? No, because the division results in a non-zero remainder of .
Think of factors as puzzle pieces that fit perfectly. When you divide one polynomial by another, a remainder of zero means it's a perfect fit—the divisor is a factor! Any other remainder means the pieces don't align correctly, so it's not a factor. This gives you a clear 'yes' or 'no' answer every single time.
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Section 1
📘 Dividing Polynomials Using Long Division
The result can be expressed as:
Next, you’ll apply this method to divide various polynomials and test if one polynomial is a factor of another.
Section 2
Polynomial long division
Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree. The final result is expressed in the form:
To divide by , we perform long division to find the quotient is with a remainder of .
To divide by , the result is a quotient of with a remainder of .
When dividing by , the quotient is with a remainder of , written as .
Think of this as the algebraic version of dividing 125 by 5. You divide the leading terms, multiply the result by the divisor, subtract the product, and bring down the next term. This process repeats until you have a remainder that is of a lower degree than the divisor, systematically simplifying complex polynomial fractions.
Section 3
Don't forget the zero placeholder
When a dividend is missing a term for a specific power of the variable, you must insert a placeholder term with a coefficient of 0. For example, to divide , you should rewrite it as .
To divide by , rewrite the dividend as to keep columns aligned.
To divide by , rewrite it as before starting the division.
Imagine lining up soldiers by rank. If a rank is missing, you leave a space for it, right? Using a zero placeholder like '' does the same thing. It keeps all your terms aligned correctly during the subtraction steps of long division, preventing you from mixing up different powers of and getting a chaotic result.
Section 4
Is it a factor? Check the remainder!
A polynomial, such as , is considered a factor of another polynomial if the remainder is 0 after division. If the remainder is any non-zero value, then it is not a factor.
Is a factor of ? Yes, because dividing gives a remainder of .
Is a factor of ? No, because the division results in a non-zero remainder of .
Think of factors as puzzle pieces that fit perfectly. When you divide one polynomial by another, a remainder of zero means it's a perfect fit—the divisor is a factor! Any other remainder means the pieces don't align correctly, so it's not a factor. This gives you a clear 'yes' or 'no' answer every single time.
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter